Modelling and Optimisation of Hysteresis and Sensitivity of Multicomponent Flexible Sensing Materials

Abstract

The development of high-performance polymeric sensing materials is urgently needed for the development of force sensors. Hysteresis and sensitivity are considered to be one of the two key metrics for evaluating the performance of force sensors, and their performance-influencing factors and optimisation models have not been addressed. In this paper, a new Kepler optimisation algorithm (HKOA) and a long short-term memory network optimisation model (HKOA-LSTM) based on HKOA are proposed, and analytical models of the hysteresis and sensitivity are derived, respectively. First, multifactor experiments were conducted to obtain experimental data for the prediction models; the prediction models for the hysteresis and sensitivity performance of sensing materials were constructed using response surface methodology (RSM), Random Forest (RF), long short-term memory (LSTM) network, and HKOA-LSTM. Next, the four prediction models were evaluated; the comparison results show that the HKOA-LSTM model performs the best. Finally, the optimal solution of the prediction model is obtained using the multi-objective RIME (MORIME) algorithm. The findings indicate a hysteresis of 3.279% and an average sensitivity of 0.046 kPa⁻¹ across a broad pressure range of 0–30 kPa when the Fe₃O₄ content is 0.665 g, the carbon nanotube (CNT) content is 1.098 g, the multilayer graphene (MLG) content is 0.99 g, and the moulding temperature (MT) is 67 °C. The simulation outcomes for the hysteresis and sensitivity closely align with the experimental test values, exhibiting relative errors of 0.765% and 0.434%, respectively. Furthermore, the sensing performance in this study shows a significant enhancement compared to prior research, with the hysteresis performance improved by 31% and sensitivity increased by 26%. This approach enhances the experimental efficiency and reduces costs. It also offers a novel strategy for the large-scale, rapid fabrication of high-performance flexible pressure sensor materials.

1. Introduction

Conductive flexible polymer materials, characterised by distinctive electrical, mechanical, and processing attributes, have garnered significant interest due to their substantial potential for flexible sensor applications, including e-skin [1,2], medical monitoring [3,4], and intelligent robotics [5,6,7]. Their properties encompass hysteresis, sensitivity, creep, repeatability, and linearity [8]. Hysteresis and sensitivity have emerged as the principal indicators influencing sensor performance, affecting reliability and measurement accuracy. In specific sensors requiring sustained high sensitivity across a broad pressure range, operations often exhibit considerable hysteresis error [9], amplifying the pressure sensor’s data processing burden and compromising sensor reliability. Consequently, developing flexible sensing materials with low hysteresis and high sensitivity is an urgent challenge that necessitates resolution.

The viscoelastic properties of the elastomer, the interfacial adhesion between the conductive material and the substrate, and the surface energy between the contacting elastomers all influence the hysteresis of the sensing material. Incorporating microstructures (e.g., porous [10], wavy [11], and pyramidal [12]) into the substrate of the flexible sensing material can significantly mitigate the hysteresis induced by the viscoelasticity of the elastomer; for instance, the hysteresis of the pollen-shaped layered sensor is merely 4.6% [13]. Elastomer microstructures are susceptible to random cracking and wear due to external stresses, leading to an increase in hysteresis and a decrease in sensor reliability [14,15]. Furthermore, researchers have examined the impact of the contact between the conductive material and the substrate on the hysteresis of the sensors. Oh et al. [16] suggested a piezoresistive pressure sensor including an elastomer template featuring evenly sized and aligned pores and a thin coating of conductive polymer chemically grafted onto the pore surfaces, exhibiting a hysteresis of merely 2%. Chen et al. [17] developed a novel double-layer MXene-doped MLG pressure sensor exhibiting a hysteresis of 1.51% within a pressure range of 0–10 kPa. Lek et al. [18] developed flexible pressure sensors utilising an MLG-based conductive foam, demonstrating remarkable hysteresis and elevated sensitivity within the low-pressure range of under 10 kPa. Guo et al. [19] developed a flexible pressure sensor through 3D printing utilising a nanocomposite material composed of multi-walled CNTs, room-temperature vulcanising (RTV) silicone rubber, and carbon nanofibers. This sensor demonstrated superior performance within a pressure range of 0–10 kPa, exhibiting a hysteresis of merely 2.44% and a sensitivity of 0.4311 kPa⁻¹. The studies mentioned above indicate that, in sensors composed of nanocomposites with nanoparticles dispersed within a polymer matrix, the interaction between the conductive polymer and the matrix significantly influences hysteresis generation. Furthermore, robust interfacial bonding between the elastic nanoparticles (e.g., CNT [19] and MLG [17,18]) and matrix effectively mitigates sensor hysteresis.

Highly sensitive flexible sensors are extensively employed for measuring the intraocular pressure and pulse, replicating the human body’s tactile perception and integrating sensing [20] and handling items. Nonetheless, an inherent trade-off exists between high sensitivity and a broad detection range [21]. Presently, the majority of tactics to surpass the sensitivity threshold of flexible pressure sensors concentrate on the fabrication of microstructures on elastomer substrates (e.g., polydimethylsiloxane [22], eco-flexible TPU [23], and polyurethane [24], among others). Nonetheless, the microstructure design process is intricate and labour-intensive, and the performance of flexible sensors with microstructures cannot be enhanced further in the low-pressure region (<20 kPa). Doping conductive materials, including carbon-based substances (e.g., carbon black [25], graphene [26], and CNTs [27]), conductive polymers (such as poly(3,4-dioxothiophene ethylene glycol) [28] and polypyrrole [29]), nitride materials (MXene [30]), or metallic materials (like silver nanoparticles [31] and nickel powders [25]), within elastomeric substrates is an effective method to enhance the sensitivity of flexible sensing materials. Doping carbonaceous nanoparticles with a high specific surface area significantly improves the electrical conductivity of composites and facilitates the establishment of conductive networks at reduced percolation thresholds [32]. Co-doping various carbonaceous nanoparticles can produce a synergistic effect, where the interaction between different materials enhances the overall performance, thereby further lowering the percolation threshold [33]. Lu et al. [34] introduced a novel technique for fabricating a flexible piezoresistive pressure sensor utilising a porous polydimethylsiloxane sponge as the substrate, incorporating reduced graphene oxide and multi-walled CNTs as fillers to create a high-performance sensor featuring a dual conducting network. The sensor features a pressure detection range of 0–200 kPa, a sensitivity of 1.62 kPa⁻¹, a reaction time of 61 ms, and can endure 22,000 loading/unloading cycles at 0–15% compressive strain. Tung [35] indicated that Fe₃O₄ nanoparticles exhibit significant chemical stability and biocompatibility, thereby preserving the stability of the conducting network within the polymer matrix. Graphene- Fe₃O₄ nanosheets were synthesised via the hydrothermal technique by Zha et al. [36]. The G-F nanosheets incorporated into the polydimethylsiloxane (PDMS) matrix significantly reduced the percolation threshold, resulting in an approximately 10 orders of magnitude enhancement in the electrical conductivity of the composite compared to that of pure PDMS. The enhanced interfacial interactions between the G-F nanosheets and the matrix also contributed to the composites demonstrating notable hysteresis. In conclusion, Fe₃O₄, CNT, and MLG nanoparticles can improve composite sensing materials’ hysteresis and sensitivity characteristics. Furthermore, there is a lack of pertinent research on the concurrent improvement of hysteresis and sensitivity in sensing materials across an extensive pressure range, rendering the investigation presented in this paper valuable.

Owing to the synergistic interactions among fillers and the intricacies of the reaction mechanism, ascertaining the appropriate quantity of these materials to incorporate remains challenging once the requisite type of filler is identified. Estimating the preparation conditions without mathematical models is a labour-intensive and expensive endeavour due to the interplay of numerous elements. Machine learning can forecast the mechanical, thermal, optical, electrical, or other characteristics of fabricated materials depending on the composition of the flexible composite and the manufacturing process [37]. Muh et al. [38] utilised RSM to predict the elastic modulus and modulus of rupture of polyvinyl alcohol/acrylonitrile/nanoclay nanocomposites. The created model had an R² value approaching 1 for the experimental data, and the residual normal probability plot conformed to a straight line, signifying a high level of accuracy in the model. Adel et al. [39] forecasted the compressive and flexural strengths of carbon nanotube (CNT)-reinforced cementitious nanocomposites via an RF model. The model accounted for 98.2% of the variability in the training data regarding compressive strength and achieved a prediction accuracy of 86.9%, demonstrating a commendable performance. The model accounted for 92.7% of the variability in the training data regarding flexural strength and achieved a prediction accuracy of 78.2%. Fathi et al. [40] examined the influence of several machining parameters (spindle speed, feed rate, and number of passes) on the mechanical properties of aluminium nanocomposites utilising a long short-term memory (LSTM) model. The model accurately predicted the ultimate tensile strength, yield strength, intrinsic frequency, and damping ratio of the material, yielding R² values of 0.912, 0.952, 0.951, and 0.987, respectively. The findings indicated that LSTM is proficient in evaluating the mechanical properties of aluminium nanocomposites. The studies mentioned above suggest that machine learning techniques, including RSM, RF, and LSTM, can proficiently forecast the mechanical properties of nanocomposites; yet, there is a paucity of research focused on predicting the electrical properties of flexible sensing materials utilising these models.

This study employs CNTs, MLGs, and magnetic nanoparticles of Fe₃O₄ as fillers, with RTV silicone rubber serving as the substrate. Flexible sensing materials are synthesised through mechanical blending, and the optimisation of hysteresis and sensitivity is accomplished using machine learning techniques. The hysteresis mechanism and sensitivity of the sensing material and its affecting elements are examined, and a matching mathematical model is developed. The Box–Behnken Design (BBD) methodology examines the impacts of four variables, Fe₃O₄, CNT, MLG nanoparticle concentration, and MT, on the hysteresis and sensitivity. Predictive models correlating these four factors with output performance are developed utilising the RSM, RF, LSTM, and HKOA-LSTM algorithms. Ultimately, the Pareto-optimal frontier solutions are identified through the MORIME algorithm, and the method’s practicality is experimentally validated.

2. Materials and Methods

2.1. Hysteresis and Sensitivity Mechanisms

2.1.1. Mechanisms of Hysteresis

Hysteresis defines the variability of the output signal during the application and release of pressure. At the same time, the resistive viscoelasticity of conductive polymer composites is influenced not only by the mechanical viscoelasticity of the matrix but also by the conductive network established through the modulation of nanoparticles in reaction to external forces [16]. This study characterises the resistive hysteresis of sensing materials by integrating the viscoelastic model with electron tunnelling theory. The multi-branch model [41] (one-dimensional generalised viscoelastic model or Prony series) expresses total stress by the equation derived from the Boltzmann superposition principle:

$$\sigma \left(t\right)={\sigma }_{\mathrm{\infty }}+{\int }_0^{t_s} \sum _{i=1}^n \dot{\epsilon }{\sigma }_i\exp \left(-{\displaystyle \frac{t-t^{\prime }}{{\tau }_i}}\right)\mathrm{d}{t}^{\prime }$$

(1)

where $\sigma \left(t\right)$ represents the stress at time t (Pa), ${\sigma }_{\mathrm{\infty }}$ denotes the equilibrium stress (Pa), ${\sigma }_i$ signifies the ith corresponding coefficient, and $\dot{\epsilon }$ indicates the constant elongation, with ${\tau }_i$ being the relaxation period of the ith branch, $t_s$ denotes the time at which the loading phase concludes, $t^{\prime }$ represents a specific time point in the past used for calculating the stress magnitude at the current moment, and $n$ signifies the number of non-equilibrium branches.

This is achieved by substituting the pressure variable in Equation (1) with the resistance value:

$$\left({\displaystyle \frac{R}{R_0}}\right)(t)=R_{\mathrm{\infty }}+\sum _{i=1}^n {\int }_0^t {\displaystyle \frac{\mathrm{d}{R}_i}{\mathrm{d}{t}^{\prime }}}\exp (-{\displaystyle \frac{t-t^{\prime }}{{\tau }_i}})\mathrm{d}{t}^{\prime }$$

(2)

In this context, $R$ and $R_0$ denote the resistance and initial resistance of the sensing material at time t, respectively (Ω); $R_{\mathrm{\infty }}$ represents the relative equivalent resistance (Ω); and $R_i$ is the ith coefficient that characterises the impact of the load on the resistance. Equation (2) may also be expressed as follows:

$$\left({\displaystyle \frac{R}{R_0}}\right)(t)=R_{\mathrm{\infty }}+\sum _{i=1}^n {\int }_0^t {\displaystyle \frac{\mathrm{d}{R}_i}{\mathrm{d}\epsilon \left(t^{\prime }\right)}}{\displaystyle \frac{\mathrm{d}\epsilon \left(t^{\prime }\right)}{\mathrm{d}{t}^{\prime }}}\exp (-{\displaystyle \frac{t-t^{\prime }}{{\tau }_i}})\mathrm{d}{t}^{\prime }$$

(3)

where $\epsilon \left(t^{\prime }\right)$ represents the strain and may be any continuous function.

Consequently, the relative resistance change $\left({\displaystyle \frac{R}{R_0}}\right)(t)$ can be articulated by the following equation:

$$\left({\displaystyle \frac{R}{R_0}}\right)(t)=R_{\mathrm{\infty }}+\sum _{i=1}^n {\int }_0^t {\displaystyle \frac{\mathrm{d}{R}_i}{\mathrm{d}\epsilon }}\dot{\epsilon }\exp \left(-{\displaystyle \frac{t-t^{\prime }}{{\tau }_i}}\right)\mathrm{d}{t}^{\prime }$$

(4)

The theory of electron tunnelling is employed to ascertain the relevant coefficients $R_i$. It is sourced from the literature [42,43]:

$$\left\{\begin{array}{c}R=\left({\displaystyle \frac{L}{N}}\right)\left({\displaystyle \frac{8\pi hs}{3a^2\gamma e^2}}\right)\exp \left(\gamma s\right)\\ \gamma ={\displaystyle \frac{4\pi (2m\varphi {)}^{0.5}}{h}}\end{array}\right.$$

(5)

$L$ represents the number of particles constituting a single conducting channel, $N$ denotes the number of conducting channels, $h$ signifies Planck’s constant, and $s$ indicates the distance between the surfaces of two adjacent particles (m). $a^2$ represents the effective cross-section for tunnelling (m²), $e$ denotes the electron charge, $m$ signifies the electron mass (kg), $\varphi$ indicates the potential barrier height between adjacent particles, and $\gamma$denotes the tunnelling decay constant in the tunnelling effect.

The alteration in electrical conductivity results from the relaxing of various segments of the polymer chains. The alteration in resistance resulting from the displacement of the ith polymer chain is as follows:

$$R_i={\displaystyle \frac{R_{i\mathrm{b}}}{R_{i\mathrm{a}}}}=\left({\displaystyle \frac{L{N}_{0}s}{L_{0}N{s}_0}}\right)\mathit{exp}\left[\gamma \right(s-s_0\left)\right]$$

(6)

$R_{i\mathrm{b}}$ represents the resistance value after loading (Ω), $R_{i\mathrm{a}}$ denotes the resistance value before loading (Ω), $L_0$ signifies the number of particles forming a single conducting path in the initial state, $N_0$ represents the number of conducting paths in the initial state, and $s_0$ indicates the distance between neighbouring particles in the initial state (m).

The separation between the surfaces of two adjacent particles subjected to an external force can be articulated as follows:

$$s=s_0(1+\dot{\epsilon }t)$$

(7)

Substituting Equation (8) into Equation (7) produces the associated coefficient of variation for resistance $R_i$.

$$\left\{\begin{array}{c}{R}_i=R_{i0}\left(1+\epsilon \right)\mathit{exp}\left(k_i\epsilon \right)\\ R_{i0}={\displaystyle \frac{L{N}_0}{L_{0}N}}\\ k_i=\gamma s_0\end{array}\right.$$

(8)

We simplify the model by treating $R_{i0}$ as a constant, resulting in the following equation:

$${\displaystyle \frac{\mathrm{d}{R}_i}{\mathrm{d}\epsilon }}=R_{i0}(1+k_i+k_i\epsilon )\mathrm{e}\mathrm{x}\mathrm{p}\left(k_i\epsilon \right)$$

(9)

Bergstrom’s experimental studies indicate that stress correlates with the logarithm of the applied strain rate across a broad spectrum of strain rates [44]. We employ a logarithmic function of the strain rate to represent the relative equilibrium resistance, which escalates with strain as indicated in Equation (9). Consequently, the equivalent resistance $R_{\mathrm{\infty }}$ is computed as follows:

$$R_{\mathrm{\infty }}=R_{\mathrm{\infty }}^0+R_{\mathrm{\infty }0}\epsilon \exp (k_{\mathrm{\infty }0}\epsilon )(1+a\mathrm{l}\mathrm{n}(1+b{\displaystyle \frac{{\int }_0^{\epsilon } \dot{\epsilon }d{\epsilon }^{\prime }}{\epsilon }}))$$

(10)

${\displaystyle \frac{{\int }_0^{\epsilon } \dot{\epsilon }d{\epsilon }^{\prime }}{\epsilon }}$ represents the mean strain rate.

Inserting Equations (9) and (10) into Equation (4) yields the relative resistance variation as a function of strain, namely, the resistive hysteresis model:

$$\begin{gathered} \left({\displaystyle \frac{R}{R_0}}\right)\left(t\right)=R_{\mathrm{\infty }}^0+R_{\mathrm{\infty }0}\epsilon \left(t\right)\mathit{exp}\left(k_{\mathrm{\infty }0}\epsilon \left(t\right)\right)\left(1+\mathit{aln}\left(1+b{\displaystyle \frac{{\int }_0^{\epsilon } \dot{\epsilon }d{\epsilon }^{\prime }}{\epsilon \left(t\right)}}\right)\right) \\ +\sum _{i=1}^n {\int }_0^t R_{i0}(1+k_i+k_i\epsilon (t^{\prime }))\mathit{exp}(k_i\epsilon (t^{\prime }))\dot{\epsilon }(t^{\prime })\mathit{exp}(-{\displaystyle \frac{t-t^{\prime }}{{\tau }_i}})d{t}^{\prime } \end{gathered}$$

(11)

The hysteresis of the composite sensing material, as indicated by Equation (11), is associated with the mechanical viscoelasticity of the rubber matrix and the adhesion between the filler and the matrix. Hysteresis is influenced by the material type, content, and fabrication process, with the filler content being the primary factor affecting the mechanical viscoelasticity of the composite material when the filler composition is fixed. Additionally, the preparation temperature significantly impacts the adhesion of the filler to the matrix. This paper examines the influence of four elements on the hysteresis of composite sensing materials: the amount of Fe₃O₄, CNT, MLG, and the material MT.

2.1.2. Mechanisms of Sensitivity

Sensitivity is a crucial property of a pressure sensor, indicating its capacity to transduce pressure into an electrical output. The slope of the pressure–resistance response curve can be intuitively visualised [45]. Sensitivity pertains to several electronic conduction theories, including field emission theory, tunnelling effect theory, and conduction route theory. This study primarily focuses on tunnelling effect theory and classical percolation theory.

The fundamental equation for the sensitivity of piezoresistive sensing materials is as follows:

$$S={\displaystyle \frac{∆R/R_0}{P}}$$

(12)

where $∆R$ represents the change in resistance, $R_0$ is the initial resistance of the composite sensing material (Ω), and $P$ is the pressure exerted on the sensing material (Pa).

The change in resistance of the sensing material presented in this paper, when subjected to pressure, is accompanied by a relatively minor and inconsequential alteration in its cross-sectional area. The relative change in resistance can be articulated using the following equation:

$${\displaystyle \frac{\mathsf{\Delta }R}{R_0}}={\displaystyle \frac{R_0-R}{R_0}}=1-{\displaystyle \frac{\rho }{{\rho }_0}}\lambda$$

(13)

In this context, $R$ denotes the resistance of the composite post-compression (Ω), ${\rho }_0$ represents the initial resistivity of the composite (Ω·m), $\rho$ indicates the resistivity of the composite following compression (Ω·m), and $\lambda$ signifies the thickness compression ratio, defined as $d/d_0$.

Classical seepage theory posits that a scaling law [46] can characterise conductivity. The initial resistance of the sensing material can be determined by the following equation:

$${\rho }_0={\displaystyle \frac{1}{\sigma }}={\displaystyle \frac{1}{{\sigma }_0({\phi }_0-{\phi }_{\mathrm{c}}{)}^t}}$$

(14)

In this context, $\sigma$ represents the composite’s conductivity (S/m), ${\sigma }_0$ denotes the scaling factor, ${\phi }_0$ signifies the filler content, ${\phi }_{\mathrm{c}}$ indicates the percolation threshold, and $t$ is the critical exponent, which is solely dependent on the dimensionality of the conducting system.

The resistivity $\rho$ can be determined using Equations (5) and (6) along with the fundamental equation for resistance:

$$\rho =R{\displaystyle \frac{S}{M}}=\left({\displaystyle \frac{SL}{MN}}\right)\left({\displaystyle \frac{8\pi hs}{3a^2\gamma e^2}}\right)\exp \left(\gamma s\right)$$

(15)

$M$ represents the length of the sensing material.

The sensitivity analytical model for composite materials can be derived by putting Equations (13)–(15) into Equation (12):

$$S={\displaystyle \frac{1-\frac{SL}{MN}\left[\frac{8\pi hs}{3a^2\gamma e^2}\exp \left(\gamma s\right)\right][{\sigma }_0({\phi }_0-{\phi }_{\mathrm{c}}{)}^t]}{E\lambda }}$$

(16)

$E$ represents the modulus of elasticity of the sensing material.

The sensitivity of the composite sensing material is intimately linked to Young’s modulus of the rubber matrix, the nanofiller’s dispersion state, and the composite’s percolation threshold, as indicated by the equation above. The nature and composition of the added fillers, together with the production process, influence these characteristics. This work will examine the sensitivity of composite sensing materials by analysing the effects of three types of nanoparticles—Fe₃O₄, CNT, and MLG—and the MT as the primary affecting factors.

2.2. Experiments

2.2.1. Materials

The composite matrix was synthesised from dimethicone oil, room-temperature vulcanised silicone rubber (RTV), and a curing agent obtained from Hong Ye-Jie Technology Co., Ltd., located at Longgang Avenue, Liuhe Community, Pingdi Street, Shenzhen, China. The Fe₃O₄ particles were supplied by China Hang-Ba Metal Materials Co., Ltd., located in Hangzhou, China, possessing an average diameter of 500 nm and a purity of 99.9 wt%. The CNTs were provided by Suzhou Tan-Feng Graphene Technology Co., Ltd., located in Yuexi Street, Suzhou, China, exhibiting a purity exceeding 95 wt%, a length larger than 3 µm, and an outer diameter ranging from 8 to 15 nm. The MLG, including 5–10 layers with an average diameter of 50 µm, was supplied by Suzhou Tan-Feng Graphene Technology Co., Ltd., located in Yuexi Street, Suzhou, China.

2.2.2. Composite Preparation

The nanocomposites were synthesised using the solution casting process, as seen in Figure 1. Initially, a specific combination of Fe₃O₄, CNTs, and MLGs was milled in a grinder, thereafter combined with an aqueous ethanol solution at a volume ratio of 3:1, and ultrasonically dispersed for 30 min. The pretreatment nanoparticles were then subjected to high-temperature drying for one hour. Subsequently, a specific quantity of coupling agent (KH550) was incorporated for surface modification treatment and placed in a 100 °C drying oven for one hour, resulting in pretreated conductive filler particles. The conductive filler was incorporated into 45 g of RTV matrix, with a 5% mass ratio of dimethyl silicone oil (DSO) for dilution. Subsequently, a stirrer was employed at a speed of 2100 r/min for 10 min, completing the mixing according to a ratio of 1.5 wt%. The curing agent (CA) was added and stirred for an additional 2 min. The mixed composite materials were then put into the mould and placed in a vacuum box for 10 min to facilitate exhaust processing, followed by further steps. The mould was subjected to a magnetic field generator with a magnetic field intensity of 300 mT for 7 min, placed in a high-degree drying oven for 3 h, and then removed to cool naturally at room temperature to yield the appropriate composite sensing material. The composition and content of the materials required for the preparation of the composites are shown in

Table 1. Material composition and content.

Name	Content (g)
Fe3O4	0.4/0.6/0.8
CNT	0.8/1/1.2
MLG	0.7/0.9/1.1
KH550	1.8
DSO	2.25
CA	0.675
RTV	45

.

2.2.3. Data Acquisition

We evaluated the hysteresis and sensitivity of the sensing materials utilising a microcomputer-controlled electronic universal testing machine (FBS200N, Forbes Testing Equipment Co., Ltd., located in Xinglong Road, Xiamen, China) and an LCR digital bridge (UC2836B, U-Tech Electronic Technology Co., Ltd., located in Pan Yang Road, Longhu Tang Street, Changzhou, China). The sample was positioned between two copper plates of the universal testing equipment, and to simultaneously measure pressure and resistance, the compression speed was established at a constant rate of 1 N/s. During the analysis of the hysteresis characteristics, the testing apparatus applied a steadily increasing uniaxial pressure on the sensing material, ranging from 0 to 30 kPa during the loading phase, followed by a uniform reduction in pressure from 30 kPa to zero during the unloading phase. The resistance values throughout the forward and reverse strokes, at identical pressure levels, were documented via the data gathering program. During the sensitivity performance analysis, the tester incrementally delivered uniaxial pressure from 0 to 30 kPa on the sensing material, halting at this pressure to record the resistance measurement.

In this study, SEM images were acquired using the Carl Zeiss GeminiSEM 360 field emission scanning electron microscope. This device, manufactured by Carl Zeiss AG, located at No. 1636, Zhi Shi Avenue, Guangzhou, China, offers high-resolution imaging and analysis capabilities for a wide range of materials. It features a resolution of 0.7 nm at 15 kV and 1.2 nm at 1 kV, with a maximum probe current of 20 nA.

2.3. Prediction and Optimisation Models

2.3.1. RSM

RSM is a fundamental regression modelling technique that employs regression and probabilistic studies to delineate an approximate and explicit functional relationship between input random variables and output responses [47]. Minimal computing work and implementing a randomised and unbiased experimental design can significantly diminish superfluous experimental labour, conserving both the time and financial resources associated with the experiment. Figure 2 illustrates the schematic diagram of the RSM model, whereas Equation (17) delineates the functional relationship between input parameters and output variables in RSM.

$$y={\alpha }_0+{\sum }_{i=1}^N{\alpha }_i{x}_i+{\sum }_{i=1}^N{\alpha }_{ii}{x}_i^2+{\sum }_{i<j}\sum {\alpha }_{ij{x}_i{x}_j}$$

(17)

In this context, $y$ represents the response variable, $x_i$ signifies the levels of the input parameter, ${\alpha }_0$ indicates the bias, ${\alpha }_i$ represents the linear effect, ${\alpha }_{ii}$ defines the squared effect of the components, and ${\alpha }_{ij}$ refers to the interaction effect between the influences.

This work employed the BBD method of RSM for the experimental design. BBD is a commonly used experimental design method in response surface methodology. It is particularly suited to situations where there is a need to explore how three or more factors affect one or more response variables. BBD uses three levels of each factor (usually denoted −1, 0, 1), which allows it to fit a quadratic model and thus capture nonlinear effects. BBD conducts the experiments at the centre level of each pair of factors, varying only two factors at a time while keeping the other factors at their centre level. This design avoids trials under extreme conditions and helps reduce costs and time. The experimental data were processed and analysed using Design-Expert software version 11.1.0.1, and the fit of the RSM model to the data was evaluated by analysis of variance (ANOVA) to ascertain the extent of influence of the independent factors on the dependent variable.

2.3.2. RF

RF is a parallel ensemble learning technique that enhances the model’s accuracy and stability by generating several decision trees and aggregating their outputs [48]. A collection of autonomous decision trees is generated utilising distinct training datasets and random attributes. Each tree is trained independently to partition the data according to different attributes and establish prediction rules, as seen in Figure 3.

In the application of RF for prediction, each tree generates a prediction, and for regression tasks, the final result is obtained by calculating the mean of these predictions. The fundamental concept of RF is that by amalgamating several decision trees, the model’s prediction bias and variance can be diminished, enhancing its accuracy and generality [49].

Table 2. RF hyperparameter ranges.

Hyperparameters	Values
Number of estimators	1–300
Maximum depth	1–300
Minimum sample split	2
Bootstrap	False
Warm start	True
Maximum features	sqrt

delineates the spectrum of values for the RF model’s public hyperparameters. This paper employed two hyperparameter optimisation strategies, GridSearchCV and RandomizedSearchCV, to enhance the RF regression model’s performance by exploring all combinations of designated hyperparameter values. The hyperparameters, precisely, the maximum tree depth and the number of decision trees, were determined to influence the model performance significantly. Subsequently, the RF predictions for each performance type are examined.

2.3.3. Long Short-Term Memory (LSTM)

The LSTM network introduced by Hochreiter and Schmidhuber is a specific recurrent neural network [50]. The structure is seen in Figure 4. The LSTM network incorporates gating units comprising input, output, and forget gates [51]. The information in memory is contingent upon the activation levels of the gating units. When the activation of the input gates is elevated, data are retained in the memory unit. Conversely, the information is transmitted to the subsequent unit if the output gate activation is high. Furthermore, the forgetting gate’s activation level dictates whether the memory cell’s information is purged. This method allows LSTM to efficiently handle and retain data over extended periods, addressing the limitations of conventional neural networks that struggle with long-term sequence issues and gradient vanishing during computation. The overarching parameter configurations of the LSTM model are presented in

Table 3. LSTM hyperparameter range.

Hyperparameters	Values
Layers	1–4
No. of neurones	1–150
Learning rate	0.001–0.01
Optimiser	Rmsprop
No. of batch size	8–32
No. of epochs	50–100
Time steps	1–20

.

2.3.4. HKOA-LSTM

The KOA algorithm, introduced by Mohamed Abdel-Basset and other researchers in 2023 [52], is a physics-based meta-heuristic inspired by Kepler’s rules of planetary motion, enabling the prediction of planetary positions and velocities at any specified moment. The algorithm employs a planet’s position, mass, gravity, and orbital velocity as the four fundamental operating parameters, with each planet’s position serving as a candidate solution and the optimal solution analogous to the sun. The algorithm particular to KOA is outlined as follows:

Step 1: Initialisation.

$$X_i^j=X_{i,\mathrm{l}\mathrm{o}\mathrm{w}}^j+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left[\mathrm{0,1}\right]\times \left(X_{i,\mathrm{u}\mathrm{p}}^j-X_{i,\mathrm{l}\mathrm{o}\mathrm{w}}^j\right)$$

(18)

$X_i$ denotes the ith planet (candidate solution) within the search space; $X_{i,\mathrm{u}\mathrm{p}}^j$ and $X_{i,\mathrm{l}\mathrm{o}\mathrm{w}}^j$ signify the upper and lower limits of the $j$th decision variable, respectively. $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d} \left[\mathrm{0,1}\right]$ refers to a randomly generated number within the interval of 0 to 1.

Step 2: Define F.

$$F_{{\mathrm{g}}_i}(t)=e_i\times \mu \left(t\right)\times {\displaystyle \frac{{\overline{M}}_s\times {\overline{m}}_{\mathrm{i}}}{{\overline{R}}_i^2+\epsilon }}+r_1$$

(19)

${\overline{M}}_s$ and ${\overline{m}}_{\mathrm{i}}$ indicate the normalised values of $M_s$ and ${\mathrm{m}}_i$, which correspond to the masses of the sun $X_s$ and the target planet $X_i$, respectively; $\epsilon$ represents a minuscule value; $\mu \left(t\right)$ is a function that exponentially diminishes over time t to regulate search precision; $e_i$ signifies the orbital eccentricity of the planet; $r_1$ is a randomly generated value within the range of 0 to 1; and ${\overline{R}}_i$ denotes the normalised value of $R_i$, representing the Euclidean distance from the sun $X_s$ to the target planet $X_i$.

Step 3: Calculate the velocity of the target planet.

$$V_i\left(t\right)=\left\{\begin{array}{l}\ell \times \left(2r_4\overrightarrow{X_i}-\overrightarrow{X_b}\right)+\stackrel{-}{\mathcal{l}}\times \left(\overrightarrow{X_a}-\overrightarrow{X_b}\right)+\left(1-R_{i-norm}\left(t\right)\right)\\ \times \Gamma \times \overrightarrow{U_1}\times \overrightarrow{r_5}\times \left(\overrightarrow{X_{i,up}}-\overrightarrow{X_{i,low}}\right), \mathrm{i}\mathrm{f}{R}_{i-norm}\left(t\right)\le 0.5\\ \\ r_4\times \mathcal{L}\times \left(\overrightarrow{X_a}-\overrightarrow{X_i}\right)+\left(1-R_{i-norm}\left(t\right)\right)\\ \times \Gamma \times U_2\times \overrightarrow{r_5}\times \left(r_3\overrightarrow{X_{i,up}}-\overrightarrow{X_{i,low}}\right), \mathrm{E}\mathrm{l}\mathrm{s}\mathrm{e}\end{array}\right.$$

(20)

$$\mathcal{l}=\overrightarrow{U}\times \mathcal{M}\times \mathcal{L}$$

(21)

$$\mathcal{L}={\left[\mu \left(t\right)\times \left(M_S+m_i\right)\left|{\displaystyle \frac{2}{R_i\left(t\right)+\epsilon }}-{\displaystyle \frac{1}{a_{i\left(t\right)}+\epsilon }}\right|\right]}^{{\displaystyle \frac{1}{2}}}$$

(22)

$$\mathcal{M}=(r_3\times (1-r_4)+r_4)$$

(23)

$$\overrightarrow{U}=\left\{\begin{array}{ccc}0& \overrightarrow{r_5}\le \overrightarrow{r_6}& \\ 1& Else& \end{array}\right.$$

(24)

$$=\left\{\begin{array}{ccc}1,& if r_4\le 0.5& \\ -1,& Else& \end{array}\right.$$

(25)

$$\stackrel{-}{\mathcal{l}}=\left(1-\overrightarrow{U}\right)\times \overrightarrow{\mathcal{M}}\times \mathcal{L}$$

(26)

$$\overrightarrow{\mathcal{M}}=\left(r_3\times \left(1-\overrightarrow{r_5}\right)+\overrightarrow{r_5}\right)$$

(27)

$${\overrightarrow{U}}_1=\left\{\begin{array}{ccc}0& \overrightarrow{r_5}\le r_4& \\ 1& Else& \end{array}\right.$$

(28)

$$U_2=\left\{\begin{array}{ccc}0& r_3\le r_4& \\ 1& Else& \end{array}\right.$$

(29)

$$a_i\left(t\right)=r_3\times {\left[T_i^2\times {\displaystyle \frac{\mu \left(t\right)\times (M_s+m_i)}{4{\pi }^2}}\right]}^{{\displaystyle \frac{1}{3}}}$$

(30)

where $V_i\left(t\right)$ represents the velocity of object $X_i$ at time t; $r_3$ and $r_4$ are randomly produced values within the region [0,1], and $\overrightarrow{r_5}$ is a vector including random values in the range [0,1]; $\overrightarrow{X_a}$ and $\overrightarrow{X_b}$ signify randomly picked solutions from the whole set of solutions; $R_i\left(t\right)$ represents the distance between the sun $X_s$ and the target planet $X_i$ at time t; $R_{i-norm}\left(t\right)$ signifies the normalised value of the Euclidean distance between the sun $X_s$ and the target planet $X_i$ at time t; $R_i\left(t\right)$ represents the distance between the best solution $X_s$ and the object $X_i$ at time t; $\overrightarrow{X_{i,up}}$ denotes the upper bound vector of the lth decision variable; $\overrightarrow{X_{i,low}}$ denotes the lower bound vector of the lth decision variable; $\overrightarrow{X_i}$ denotes the position vector of the lth planet in the search space; and $T_i$ represents the orbital period of object i.

Step 4: Revise the planet’s location and its distance from the sun.

$${\overrightarrow{X}}_i\left(t+1\right)={\overrightarrow{X}}_i\left(t\right)+\Gamma \times {\overrightarrow{V}}_i\left(t\right)+\left(F_{g_i}\left(t\right)+\left|r\right|\right)\times \overrightarrow{U}\times \left({\overrightarrow{X}}_s\left(t\right)-{\overrightarrow{X}}_i\left(t\right)\right)$$

(31)

$r$ is a randomly generated value within the range of 0 to 1.

$$\begin{gathered} {\overrightarrow{X}}_1\left(t+1\right)={\overrightarrow{X}}_1\left(t\right)\times {\overrightarrow{U}}_1+\left(1-{\overrightarrow{U}}_1\right)\times \\ \left({\displaystyle \frac{{\overrightarrow{X}}_i\left(t\right)+{\overrightarrow{X}}_s+{\overrightarrow{X}}_a\left(t\right)}{3.0}}+\left({\displaystyle \frac{{\overrightarrow{X}}_i\left(t\right)+{\overrightarrow{X}}_s+{\overrightarrow{X}}_a\left(t\right)}{3.0}}-{\overrightarrow{X}}_b\left(t\right)\right)\right) \end{gathered}$$

(32)

${\overrightarrow{X}}_i\left(t+1\right)$ denotes the updated position of the target planet $X_i$ at time $t+1$; ${\overrightarrow{V}}_i\left(t\right)$ represents the velocity necessary for planet $X_i$ to attain the new position; and ${\overrightarrow{X}}_s\left(t\right)$ indicates the optimal position of the sun identified thus far.

Step 5: Elite strategy.

$${\overrightarrow{X}}_{i,n}(t+1)=\left\{\begin{array}{l}{\overrightarrow{X}}_i(t+1),iff\left({\overrightarrow{X}}_i\right(t+1\left)\right)\le f\left({\overrightarrow{X}}_i\right(t\left)\right)\\ {\overrightarrow{X}}_i\left(t\right) Else\end{array}\right.$$

(33)

Nevertheless, in the context of high-dimensional complex applications, traditional KOA is currently hindered by sluggish convergence rates, an imbalance between exploration and exploitation, inadequate convergence precision, a tendency to become trapped in locally optimal solutions, and the necessity for multiple function evaluations [53]. This paper proposes a ranking-based update mechanism (RUM) to exclude solutions that do not yield superior results over several consecutive iterations, replacing them with new solutions generated by the update scheme. This mechanism enhances the model’s convergence speed while preventing entrapment in local optima. The technique has two stages. The first involves borrowing a fish aggregation device (FAD) from a marine protected area (MPA) to aid the operator’s exploration. The mathematical formulation of the FAD is shown in the following equation [54]:

$${\overrightarrow{\mathrm{X}}}_{\mathrm{i}}=\left\{\begin{array}{l}{\overrightarrow{\mathrm{X}}}_{\mathrm{i}}+CF\times \left[{\overrightarrow{X}}_{i,low}+r_2\times \left({\overrightarrow{X}}_{i,up}-{\overrightarrow{X}}_{i,low}\right)\right]\otimes \overrightarrow{\mathrm{U}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}if r<FADs\\ {\overrightarrow{\mathrm{X}}}_{\mathrm{i}}+[FADs\times (1-r)+r]({\overrightarrow{\mathrm{X}}}_{\mathrm{r}1}-{\overrightarrow{\mathrm{X}}}_{\mathrm{r}2})\hspace{0.25em}Else\end{array}\right.$$

(34)

${\overrightarrow{\mathrm{X}}}_{\mathrm{r}1}$ and ${\overrightarrow{\mathrm{X}}}_{\mathrm{r}2}$ represent two randomly selected individuals, $r_2$ is a random variable within the interval (0,1), and FADs are fixed at 0.2 in the MPA. Still, they are randomly generated between 0 and 1 in this context to simplify parameter tuning, $\otimes$ denotes the element-wise multiplication operator, $\overrightarrow{\mathrm{U}}$ is the binary vector utilised to determine whether to update the dimension in the current solution, and CF is the adaptive control parameter, generated at each iteration by

$$CF=(1-{\displaystyle \frac{t}{T_{max}}}{)}^{\left(2{\displaystyle \frac{t}{T_{max}}}\right)}$$

(35)

$T_{max}$ denotes the maximum number of iterations.

Secondly, an update program is intended to further investigate the search space to identify regions that may harbour solutions. This updating mechanism utilises Lévy flight and the normal distribution to provide various step sizes, hence maximising coverage of the search space. The mathematical expression of the subsequent step is as follows:

$${\overrightarrow{\mathrm{X}}}_{\mathrm{i}}=\left\{\begin{array}{l}{\overrightarrow{\mathrm{X}}}_{\mathrm{i}}+RL\times \left(\mathrm{R}\mathrm{L}\times {\overrightarrow{\mathrm{X}}}_s\left(t\right)-{\overrightarrow{\mathrm{X}}}_{\mathrm{i}}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}if r<r_2\\ {\overrightarrow{\mathrm{X}}}_{\mathrm{a}}+rn\times \left(\mathrm{r}\mathrm{n}\times {\overrightarrow{\mathrm{X}}}_b\left(t\right)-{\overrightarrow{\mathrm{X}}}_{\mathrm{r}1}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}Else\end{array}\right.$$

(36)

where $RL$ denotes the value produced by the Lévy flight, and $rn$ is a randomly generated value in the interval (0,1). Our suggested algorithm incorporates a stochastic trade-off between the first and second phases, as demonstrated in the subsequent equation:

$${\overrightarrow{\mathrm{X}}}_{\mathrm{i}}=\left\{\begin{array}{l}Eq.\left(34\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}if r\le r_2\\ Eq.\left(36\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}Else\end{array}\right.$$

(37)

Before initiating the optimisation process, a variable is created for each solution to document the count of successive failures in identifying a superior option. If the variable’s value for a solution exceeds a threshold, that solution is substituted with a new solution given by Equation (37).

The strategy above enhances the exploration model of KOA; nonetheless, its utilisation operator requires more refinement to achieve superior solutions with fewer function evaluations. This work proposes a novel strategy that employs an enhancement approach linking the updating process of the present solution to the optimal solution previously attained, aiming to identify a superior solution expeditiously. The mathematical representation of this method is illustrated in the subsequent equation:

$${\overrightarrow{\mathrm{X}}}_{\mathrm{i}}=\left\{\begin{array}{l}{\overrightarrow{\mathrm{X}}}_{\mathrm{i}}\otimes {\overrightarrow{\mathrm{U}}}_1+\left({\displaystyle \frac{\left({\overrightarrow{\mathrm{X}}}_{\mathrm{i}}+{\overrightarrow{\mathrm{X}}}_{\mathrm{S}}\left(t\right)\right)}{2}}+\left|\mathrm{r}\mathrm{n}\right|\times \left({\displaystyle \frac{\left({\overrightarrow{\mathrm{X}}}_{\mathrm{i}}+{\overrightarrow{\mathrm{X}}}_{\mathrm{s}}\left(t\right)\right)}{2}}-{\overrightarrow{\mathrm{X}}}_{\mathrm{i}}\right)\right)\otimes \left(1-{\overrightarrow{\mathrm{U}}}_1\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}if r<r_2\\ {\overrightarrow{X}}_S(t)+{\mathrm{r}}_1\times \left({\overrightarrow{X}}_S\left(t\right)-{\overrightarrow{X}}_i\left(t\right)\right)+(1-{\mathrm{r}}_1)\times \left({\overrightarrow{X}}_a\left(t\right)-{\overrightarrow{X}}_b\left(t\right)\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}Else\end{array}\right.$$

(38)

where ${\mathrm{r}}_1$ is a random number in the interval (0,1); ${\overrightarrow{X}}_S\left(t\right)$ represents the position vector of the current global optimal solution; ${\overrightarrow{\mathrm{U}}}_1$ is a binary vector designated as 1 or 0 based on the estimated convergence rate factor ($CR$); and ${\overrightarrow{\mathrm{U}}}_1$ is formally defined as follows:

$$\overrightarrow{{\mathrm{U}}_1}=\left\{\begin{array}{cc}1& if\overrightarrow{\mathrm{R}}<CR\\ 0& Else\end{array}\right.$$

(39)

The strategy and the RUM are effectively integrated with the traditional KOA to augment its search capabilities. The traditional KOA is predominantly utilised at the onset of the optimisation process to encompass the area of the search space that may harbour the desired answer. To enhance the exploration operator of the KOA, RUM is utilised to substitute inferior answers in succeeding iterations. As the quantity of function evaluations escalates, HKOA incorporates an enhancement approach to thoroughly leverage the vicinity surrounding the identified optimal solution to discover a superior answer. This optimisation process is reiterated until the termination criterion is satisfied. This study employs HKOA to determine the ideal hyperparameters of the LSTM algorithm, including the time step, number of training rounds, batch size, and learning rate, with the HKOA-LSTM flowchart illustrated in Figure 5.

The issue of overfitting must be mitigated during the training of LSTM models, which arises when the model’s complexity increases excessively due to an abundance of parameters. This complexity results in duplicate information during the model fitting process, diminishing its capacity to generalise to new data. Although augmenting the training dataset may alleviate overfitting, acquiring such data through experiments can incur substantial expenses. Consequently, creating models that can proficiently generalise predictions with minimal data is essential. To tackle this difficulty, we improved the primary hyperparameters of the LSTM model (learning rate, time step, batch size, and number of training epochs) via HKOA, and the optimal hyperparameters are presented in

Table 4. HKOA-LSTM hyperparameter values.

Hyperparameters	Values
Layers	1–4
No. of neurones	1–150
Learning rate	0.001
Optimiser	Rmsprop
No. of batch size	10
No. of epochs	65
Time steps	11

.

2.3.5. MORIME

In 2023, Su et al. [55] introduced an optimisation algorithm named RIME. The algorithm was inspired by the natural process of RIME formation and devised a soft RIME search method, a hard RIME perforation mechanism, and a forward greedy selection technique. These tactics collaboratively enhance the optimisation process, endowing the RIME algorithm with robust global optimisation capabilities. The MORIME algorithm [56] subsequently integrates the non-dominated sorting technique with the diversity-preserving congestion distance approach derived from the NSGA-II algorithm. The algorithm initially uses the non-dominated ordering technique to categorise the population into distinct non-dominated classes, subsequently computes the crowding distance for each class, and ultimately applies the crowding distance method to maintain the variety of the optimal solution set. Figure 6 illustrates the flowchart of the MORIME algorithm.

2.4. Assessment Metrics

The assessment metrics for general predictive models encompass the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE), represented by the following formulas:

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n {\left(Y_i-{Y_{pred}}_i\right)}^2}{\sum _{i=1}^n {\left(Y_i-\overline{Y}\right)}^2}}$$

(40)

$$MSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N {\left(Y_i-{Y_{pred}}_i\right)}^2}$$

(41)

$$MAE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|Y_i-{Y_{pred}}_i\right|$$

(42)

R² is a statistic that measures the goodness of fit of a regression model, which indicates the proportion of the total variance explained by the model. R² on the training set mainly reflects the model’s ability to learn from known data. In contrast, R² on the test set focuses more on the model’s generalisation ability, i.e., the model’s ability to handle unknown data.

3. Results

3.1. Evaluation of Predictive Models

To select the most precise prediction model for the hysteresis and sensitivity of the sensing material, we analysed the correlation coefficients of the four models, as presented in

Table 5. R² values for training and test datasets of various predictive models regarding hysteresis and sensitivity performance.

	Determination Coefficient	RSM	RF	LSTM	HKOA-LSTM
Hysteresis	R2 training	-	1.0000	0.9987	1.0000
	R2 testing	0.8961	0.8277	0.8864	0.9810
Sensitivity	R2 training	-	0.9999	0.9968	1.0000
	R2 testing	0.8955	0.8046	0.8440	0.9803

. The optimal prediction model for the hysteresis of the sensing material is HKOA-LSTM, with three hidden layers with 80 neurones each, achieving an R² of 0.981. The HKOA-LSTM model, featuring three hidden layers and 70 neurones per layer, has the highest sensitivity prediction accuracy, with an R² value of 0.9803, indicating superior accuracy and generalisation relative to the other three models. The other three models exhibit superior accuracy and generality. Our study reveals that the sensing properties of nanocomposites exhibit complexity, influenced by the three filler contents and MT examined, which affect the hysteresis and sensitivity of the composite sensing materials. This complexity arises from factors such as the adhesion between the matrix and filler, the interaction among filler particles, the modulus of elasticity of the matrix, and the degree of dispersion of nanofillers. Furthermore, the proposed HKOA-LSTM prediction model demonstrates a strong adaptability to the intricacies of the data. Figure 7 depicts the correlation between the predicted and measured values of the hysteresis and sensitivity characteristics of the examined composite sensing materials. Ten simulations were conducted under identical optimal conditions to evaluate the reproducibility of the HKOA-LSTM model. The graphs demonstrate a strong correlation between the projected values and the observed data since the points are closely aligned with the diagonal line, indicating an excellent model fit. The slope of the graph, consistently approaching 1 in all instances, further substantiates this observation.

To evaluate the precision of the HKOA-LSTM prediction model presented in this paper, the root mean square error (RMSE) and mean absolute error (MAE) of the four models were compared, with the results displayed in

Table 6. Comparative analysis of RMSE and MAE for four predictive models regarding hysteresis and sensitivity.

	Metrics	RSM	RF	LSTM	HKOA-LSTM
Hysteresis	RMSE	0.0865	0.1354	0.0521	0.0225
	MAE	0.0189	0.0852	0.0079	0.0094
Sensitivity	RMSE	0.2820	0.3600	0.1865	0.1285
	MAE	0.1030	0.2100	0.0994	0.0547

. The HKOA-LSTM model exhibits the lowest RMSE (0.0225) for hysteresis, signifying a superior predictive accuracy relative to that of the other models. Nonetheless, for MAE, the LSTM model exhibits a slightly lower value (0.0079), indicating that the average discrepancy between its projected and experimental values is reduced. The HKOA-LSTM model exhibits the lowest RMSE (0.1285) and MAE (0.0547) for sensitivity, signifying that its results align more with the experimental values. Regarding the hysteresis and sensitivity of the sensing material, the HKOA-LSTM demonstrates superior data fitting and prediction outcomes relative to the other three predictive models.

3.2. Optimisation of the Sensing Material Production Process via the MORIME Algorithm

The multi-objective optimisation technique aims to achieve material ratios and MTs that align with superior hysteresis and sensitivity characteristics. The examination of the experimental data presented in Table 1 indicates that the hysteresis and sensitivity of the composite sensing material exhibit a mutually exclusive relationship. To reconcile the two optimisation objectives of hysteresis and sensitivity, the HKOA-LSTM model was formulated as the objective function of the MORIME algorithm, with the concentrations of the three types of nanofillers and the MT serving as constraints. The resultant findings are illustrated in Figure 8, depicting the fabrication process of the sensing material. The factors are interrelated and exhibit a competitive dynamic; specifically, the hysteresis progressively intensifies as sensitivity increases. The Pareto-optimal solution O was identified utilising the CRITIC approach, yielding model simulation findings of hysteresis = 3.279% and sensitivity = 1.382 kPa⁻¹. The related model parameters were Fe₃O₄: 0.665 g, CNTs: 1.098 g, MLGs: 0.99 g, and MT: 67 °C.

3.3. Experimental Validation and Performance Testing of Sensing Materials

We employed the optimised results as experimental parameters for synthesising the sensing material to validate the plausibility of the model’s predictive outcomes. Figure 9a,b illustrate the produced sensing material’s hysteresis and sensitivity performance plots. Figure 9a illustrates the hysteresis performance of the sensor within a pressure range of 0–30 kPa, yielding a computed value of 3.254%. The slopes of the segments of the fitted straight line in Figure 9b represent the sensitivities of the sensors at the respective pressures. At low pressures (<1 kPa), the sensor has a sensitivity of 2.556 kPa⁻¹, with deformation occurring immediately upon applying force, reaching its maximum extent. The sensor’s sensitivity in the medium pressure range (1–5 kPa) is 1.179 kPa⁻¹. In the high-pressure range (5–9 kPa), the sensitivity of the sensor is 0.039 kPa⁻¹; when the pressure is increased to a certain extent, due to the existence of the compression limit of the composite material, at this time, the material deformation decreases. The contact area and the number of the conductive particles inside the composite material decreases, leading to a decrease in the change in the electrical resistance, which is less sensitive at this time. In the high-pressure range (9–30 kPa), the sensitivity of the sensor is 0.0054 kPa⁻¹; when the applied pressure makes the composite material enter the inelastic deformation stage, the molecular structure inside the composite material will be irreversibly deformed and at this time, the conductive path inside the material reaches a saturated state, which makes the resistance change insignificant. The sensitivity of the sensor in this state is extremely low. The data analysis reveals a significant performance advantage of the sensing material produced under optimum experimental circumstances, ensuring little hysteresis and good sensitivity across a broad pressure range of 0–30 kPa.

Table 7. Comparative analysis of hysteresis and sensitivity in this study versus those in other studies.

Ref.	Materials	Hysteresis (%)	Sensitivity (kPa−1)	Range (kPa)	Time (ms)
[57]	PDMS	4.42	3.73	<100	<50
[58]	CNTs Silver nanowire	7.12	3.179	<2	<100
[59]	Graphene CNTs	6.34	0.02	<6.5	-
[60]	Single-layer graphene	4.2	0.043 (2~10 kPa)	<10	5000
[61]	Hybridised graphene	7.5	0.032 (0~0.7 kPa)	<0.7	-
[62]	Bilayer graphene	-	0.122 (0~5 kPa) 0.077 (5~20 kPa)	<20	70
This work	Fe3O4/Graphene CNTs	3.254	2.556 (0~1 kPa) 1.179 (1~5 kPa) 0.0054 (5~30 kPa)	<30	<370

presents a comparative analysis of the hysteresis and sensitivity performance of the composite sensing materials developed in this investigation against previous flexible pressure sensors. Relative to the literature [57,58], the sensing materials in this study demonstrate an inferior sensitivity performance but an improved hysteresis performance. In comparison to the literature [59], the sensing materials examined in this study exhibit exceptional performance and an expanded pressure range, attributable to the high chemical stability and biocompatibility of Fe₃O₄ nanoparticles, which diminish the seepage threshold of the composite material and enhance conductivity. Compared to the literature [60,61,62], the flexible sensing materials produced at this institute exhibit a superior hysteresis and sensitivity performance within the same pressure range. The substantial specific surface area of the CNTs facilitates the adsorption of two graphene molecules per CNT molecule, resulting in a three-dimensional mesh structure that amplifies the contact area with the RTV substrate, thereby improving the overall conductivity of the sensing material and enhancing its sensitivity. Furthermore, the robust interaction between the CNT, MLG, and the substrate contributes to the superior hysteresis performance. The overall performance of the sensing materials created in this article is superior, and the fabrication procedure is more straightforward, further illustrating the efficacy of integrating the HKOA-LSTM method for optimising the performance of composite sensing materials.

To evaluate the applicability of the sensors in practical situations, we assessed additional characteristics of the sensing materials under investigation, including the reaction time, repeatability, stability, and durability. We employed a push/pull tester and a digital multimeter to evaluate the sensing material’s reaction time more precisely. Given that the force application duration of the tester is exceedingly brief (<0.5 ms), significantly shorter than the output response time of the digital multimeter, it can be disregarded; thus, the digital multimeter’s response time is considered the sensor’s reaction time. Figure 10a–c illustrate that the sensor’s reaction time and recovery time are 0.21 s and 0.37 s, respectively, which are enough for practical applications. The durability and stability of the proposed pressure sensing materials were evaluated through loading–unloading force cycles, as depicted in Figure 10d. After 2000 cycles, there was negligible attenuation of the resistance signals, indicating high stability and durability.

4. Discussion

4.1. Internal Structure of Flexible Sensing Materials

In Figure 11a, the MLG sheets have a distinct lamellar structure with curling at the edges. Figure 11b illustrates that the robust intermolecular van der Waals forces and the elevated specific surface area of the CNTs resulted in the significant agglomeration and intertwining of the surface-modified CNTs lacking the surface modification treatment. In Figure 11c, the Fe₃O₄ particles were effectively affixed to the MLGs. Figure 11d illustrates that the dispersion of the CNTs was markedly enhanced following the modification treatment with the coupling agent KH550, and the incorporation of the MLGs and Fe₃O₄s not only effectively mitigated the agglomeration of the CNTs but also facilitated improved interaction among the fillers.

4.2. Experiments on Electrical Properties of Flexible Sensing Materials

According to the outcomes of the one-factor experiment, level coding was performed by centring the point with the maximum effect (0) as the centroid to ascertain the experimental range of each influencing factor. The specific level factors for each influence factor are shown in

Table 8. Levels and coding of four factors.

Code	Level Factor
	${\mathbf{Fe}}_3{\mathbf{O}}_4 (\mathbf{g}$)	$\mathbf{CNT} (\mathbf{g}$)	$\mathbf{MLG} (\mathbf{g}$)	MT (°C)
Low (−1)	0.4	0.8	0.7	40
Central (0)	0.6	1	0.9	60
High (1)	0.8	1.2	1.1	80

. The detailed BBD multifactorial experimental technique is presented in

Table 9. A multifactorial experimental program was developed using the BBD approach.

Run	Influencing Factors				Hysteresis (%)	Sensitivity (kPa−1)
	Fe3O4	CNT	MLG	MT
1	1	1	0	0	5.68	0.982
2	0	1	−1	0	5.556	0.848
3	0	−1	0	1	11.654	0.614
4	−1	0	0	1	11.594	0.904
5 *	0	0	0	0	6.165	0.985
6	1	−1	0	0	6.131	0.761
7 *	0	0	0	0	5.551	0.978
8	1	0	0	1	9.541	0.838
9	−1	0	1	0	7.343	1
10	−1	0	0	−1	7.63	0.829
11	−1	0	−1	0	9.23	0.742
12	0	0	1	−1	8.874	0.879
13 *	0	0	0	0	5.683	0.922
14	0	1	0	1	12.623	0.852
15	1	0	−1	0	5.211	1.077
16	1	0	1	0	6.621	0.865
17	0	−1	0	−1	8.31	0.507
18	0	1	0	−1	6.356	0.876
19	−1	1	0	0	6.854	0.926
20 *	0	0	0	0	6.34	0.927
21	0	0	−1	−1	5.7	0.764
22	1	0	0	−1	5.015	1.059
23	0	−1	−1	0	9.526	0.631
24 *	0	0	0	0	6.111	0.891
25	−1	−1	0	0	7.884	0.663
26	0	1	1	0	9.1	0.889
27	0	−1	1	0	6.188	0.628
28	0	0	−1	1	13.536	0.799
29	0	0	1	1	11.481	0.856

* indicates the central repeated trials.

.

4.3. Forecasting Electrical Characteristics of Flexible Sensing Materials by Regression Modelling

This work employed four distinct models, RSM, RF, LSTM, and HKOA-LSTM, to forecast the electrical properties of nanocomposites. These four models were selected for their capacity to manage nonlinear data, generalise pattern recognition, and enhance prediction accuracy. A proportion of 75% of the dataset acquired from the experiment was allocated for training, while 25% was designated for testing. Identifying the optimal hyperparameter setup is essential throughout the training phase. In neural networks, hyperparameters are predetermined values chosen before model execution, significantly influencing performance.

The subsequent sections will provide a detailed discussion of each of the four forecasting models.

4.3.1. Hysteresis

RSM

The ANOVA findings for the hysteresis performance response model of multicomponent sensing composites are presented in

Table 10. Analysis of variance for hysteresis response.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value	Status
Model	29.48	14	4.25	26.81	<0.0001	Significant
$x_1$-Fe3O4	1.5264	1	0.0002	75.632	<0.0001
$x_2$-CNT	6.829	1	26.89	6.86	<0.0001
$x_3$-MLG	1.70	1	1.70	1.74	0.0055
$x_4$-MT	0.2506	1	0.2506	125.58	0.0229
$x_1{x}_2$	0.3147	1	0.3147	1.99	0.1806
$x_1{x}_3$	3.59	1	3.59	22.62	0.0003
$x_1{x}_4$	3.36	1	3.36	21.23	0.0004
$x_2{x}_3$	0.4212	1	0.4212	2.66	0.1253
$x_2{x}_4$	0.1129	1	0.1129	0.7124	0.4128
$x_3{x}_4$	0.0346	1	0.0346	0.2183	0.6475
${x_1}^2$	0.0954	1	0.0954	0.6018	0.4508
${x_2}^2$	2.075	1	20.75	130.91	<0.0001
${x_3}^2$	2.18	1	2.18	13.78	0.0023
${x_4}^2$	3.72	1	3.72	23.48	0.0003
Residual	2.22	14	0.1585
Lack of Fit	1.94	10	0.1940	2.79	0.786	Not significant
Pure Error	0.2781	4	0.0695
Cor Total	31.69	28
$R^2$	0.8961
$R_{Adj}^2$	0.8891
${\mathrm{R}}_{\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{d}}^2$	0.8633
C.V%	3.53
Adeq Precision	28.1654

. The regression model exhibits an F-value of 26.81, with the relevance of the influencing factors ranked as follows: MT > Fe₃O₄ > CNT > MLG. For quadratic terms, the ordering is $x_2^2$ > $x_4^2$ > $x_1{x}_3$ > $x_1{x}_4$. A higher F-value indicates a more substantial impact of the factors on the response value. The p-value for the model term is below 0.0001, signifying its significance; conversely, the p-value for the lack of fit term is 0.786, exceeding 0.05, indicating its insignificance. The R² of the RSM model is 0.8961, demonstrating a high degree of fit. The coefficient of variance (C.V%) is 3.53%, below 5%, suggesting that the model possesses good reliability and accurately represents the experimental data.

Figure 12a,b illustrate the influence of the Fe₃O₄ and carbon-based conductive fillers on the hysteresis of the flexible sensing material, with the variation in the hysteresis response value exhibiting relative smoothness due to the interaction of many nanoparticles. The hysteresis of the sensing material exhibits a gradual decline as the concentrations of Fe₃O₄, CNT, and MLG increase. Notably, when carbon particles interact with Fe₃O₄, the hysteresis variation becomes comparatively more stable, aligning with the findings in the literature [63], which indicate that Fe₃O₄ particles positively influence the sensing material’s ability to sustain a rapid response. Figure 12c,e,f illustrate the interaction between the filler and the MT, indicating that the hysteresis of the sensing material generally increases with temperature throughout the moulding process. This is supported by Equation (11), suggesting that an elevation in the MT results in a modification of the viscoelasticity of the filler matrix, which subsequently influences the mechanical properties of the sensing unit and causes an increase in hysteresis.

According to the ANOVA regression coefficient of BBD, the response relationship of hysteresis with the filler content and moulding temperature can be expressed as follows:

$$\begin{gathered} y_1=3.01148-5.48927x_1+1.62977x_2-2.52130x_3+0.019542x_4\dots \\ +9.27250x_1{x}_2+9.17000x_1{x}_3-0.038762x_1{x}_4\dots \\ -4.97750x_2{x}_3+0.008225x_2{x}_4+0.010900x_3{x}_4\dots \\ -4.71108x_1^2-2.99858x_2^2-0.782333x_3^2-0.000046x_4^2 \end{gathered}$$

(43)

where $y_1$ denotes the sensor hysteresis performance response value; $x_1$, $x_2$, and $x_3$ denote the content of Fe₃O₄, CNT, and MLG, respectively; and $x_4$ is the moulding temperature.

RF

The model’s hysteresis prediction outcomes are displayed in

Table 11. R² values using various hyperparameters and search methodologies in forecasting hysteresis utilising RF.

Search Techniques	Number of Estimators	Maximum Depth	R2 Train	R2 Test
GridSearchCV	100	200	1.000	0.8277
	100	20	0.999	0.8135
	200	20	0.996	0.7923
RandomizedSearchCV	42	71	0.997	0.8153
	10	94	0.999	0.7216
	102	94	1.000	0.7897

. Augmenting the quantity of decision trees does not enhance the model’s performance on the test set. For instance, when comparing models with 100 and 200 decision trees using GridSearchCV, the former exhibits a marginally superior correlation coefficient R² (0.8277 compared to 0.7923). The RF model, which includes 42 decision trees and utilises the RandomizedSearchCV approach, exhibits a superior accuracy compared to that of the model with 102 decision trees (0.8153 vs. 0.7897). The model’s performance on the test set generally diminishes as the maximum depth of the trees is reduced. For instance, employing GridSearchCV, the model with a maximum depth of 200 achieves an R² value of 0.8277 on the test set, whereas the model with a maximum depth of 20 attains an R² of 0.8135 with the same number of decision trees. The RandomisedSearchCV method indicates that the parameter combination with a maximum depth of 71 yields superior accuracy compared to a maximum depth of 94 (0.8153 vs. 0.7897). This combination also results in a higher R², implying that an increased maximum depth enables the model to comprehend more intricate relationships within the data, thereby enhancing the predictive performance. The RF model, utilising a hyperparameter configuration of 100 decision trees and a maximum depth of 200, attained the greatest R² values (1 and 0.8277, respectively) in both the training and test set through the GridSearchCV method.

LSTM

Figure 13a,b illustrate the performance of the LSTM in terms of the number of hidden layers and the number of neurones per layer, respectively. In Figure 13a, the R² value of the model on the training set rises with the addition of hidden layers. This suggests that increasing the number of hidden layers enhances the model’s ability to match the training data and capture more intricate relationships. The R² reaches its maximum with three hidden layers in the test data, signifying that the model enhances its generalisation capacity by extracting data features for improved prediction. The addition of a fourth hidden layer results in a decline in the LSTM’s predictive performance on the test dataset, signifying that the model becomes excessively complex and begins to overfit the training data, hence impairing its generalisation capability. Consequently, we established the optimal quantity of hidden layers for the model at three. The subsequent analysis focused on the amount of neurones within the hidden layers. As illustrated in Figure 13b, the R² values for the training and test datasets varied with the number of neurones. While no definitive trend emerged, the optimal model performance was achieved with 90 neurones in each of the three hidden layers, resulting in R² values of 0.9987 for the training set and 0.8864 for the test set. Incorporating additional neurones beyond the optimal threshold results in overfitting and a reduction in model generalisation. The model may overfit the details and noise in the training data rather than identifying more pertinent features, resulting in diminished performance on the dataset.

HKOA-LSTM

In Figure 14a, the model’s superior performance on the training set remains nearly the same as the number of hidden layers in the neural network varies. The R² of the model on the test set rises with the addition of hidden layers, peaks at three hidden layers, and subsequently declines upon expanding to four layers. Consequently, we select a network architecture with three hidden layers. Figure 14b examines the impact of the neurone count in these three hidden layers on model performance. As the quantity of neurones in the hidden layers escalates, the model’s performance remains exceptional and nearly invariant within the training set. In the test set, R² progressively rises with an increase in the number of neurones in the hidden layer, achieving optimal model performance at 80 neurones. A compromise between the model tunability and generalisation capability is attained, resulting in R² values of 1 and 0.981 for the training and test sets, respectively.

4.3.2. Sensitivity

RSM

Table 12. Analysis of variance for sensitivity response.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value	Status
Model	0.5096	14	0.0364	55.80	<0.0001	Significant
$x_1$-Fe3O4	0.0224	1	0.0224	15.85	0.0014
$x_2$- CNT	0.2051	1	0.2051	145.40	<0.0001
$x_3$- MLG	0.0055	1	0.0055	3.87	0.0693
$x_4$-MT	0.0002	1	0.0002	0.1536	0.7010
$x_1{x}_2$	0.0004	1	0.0004	0.3126	0.5849
$x_1{x}_3$	0.0552	1	0.0552	39.14	<0.0001
$x_1{x}_4$	0.0219	1	0.0219	15.52	0.0015
$x_2{x}_3$	0.0005	1	0.0005	0.3430	0.5674
$x_2{x}_4$	0.0043	1	0.0043	3.04	0.1031
$x_3{x}_4$	0.0008	1	0.0008	0.5961	0.4529
${x_1}^2$	0.0084	1	0.0084	5.93	0.0289
${x_2}^2$	0.1415	1	0.1415	100.32	<0.0001
${x_3}^2$	0.0147	1	0.0147	10.41	0.0061
${x_4}^2$	0.0343	1	0.0343	24.31	0.0002
Residual	0.0198	14	0.0014
Lack of Fit	0.0134	10	0.0013	0.8421	0.6262	Not significant
Pure Error	0.0064	4	0.0016
Cor Total	0.5294	28
$R^2$	0.8955
$R_{Adj}^2$	0.8712
${\mathrm{R}}_{\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{d}}^2$	0.897
C.V%	1.28
Adeq Precision	25.36

presents the ANOVA results regarding the sensitivity of the flexible sensing materials. The regression model’s total F-value is 55.80, with the affecting components ranked by importance as CNT > Fe₃O₄ > MLG > MT. For quadratic terms, the ordering is $x_2^2$ > $x_1{x}_3$ > $x_4^2$ > $x_1{x}_4$. The p-value for the model term is below 0.0001, signifying its significance; conversely, the p-value for the lack of fit term is 0.6262, over 0.05, indicating its insignificance. The R² value of the RSM model is 0.8955, signifying a strong fit of the model. The regression model’s coefficient of variation (C.V.%) is 1.28%, below 5%, meaning that the model demonstrates strong reliability and accuracy.

Figure 15a,b illustrate the influence of Fe₃O₄ particles and carbon conductive filler on sensor sensitivity. The sensitivity response exhibits considerable fluctuations due to the interaction between the three types of nanoparticles, underscoring the significant role of carbon particles in enhancing sensitivity. The carbon material possesses excellent electrical conductivity and flexibility, and their combination facilitates the formation of a carbon conductive network, which further improves dispersion, increases the contact area between the filler and substrate, enhances the conductivity of the flexible sensing material, and aligns with the conclusion of Equation (16). Data analysis reveals that as the concentrations of Fe₃O₄, CNT, and MLG increase, the sensitivity rises before declining. This phenomenon occurs due to the densification of the conductive network with the addition of conductive fillers, resulting in a reduction in the equivalent micro-resistance within the flexible sensing material. Consequently, the sensitivity diminishes, highlighting the significant influence of optimal filler content proportions on sensitivity performance. Figure 15c,e,f illustrate the interaction between the filler and the MT, demonstrating an initial increase in sensitivity with rising temperature, followed by a slow decline. Increased temperature results in the softening of the matrix, thereby reducing the average spacing between the conductive fillers during material formation, which amplifies the synergistic effect among the fillers. As the temperature rises, the matrix becomes increasingly softened, leading to a continual decrease in the spacing between the conductive fillers. This ultimately causes the agglomeration of conductive particles, disrupting the internal conductive network (a three-dimensional system of conductive pathways formed by dispersed conductive particles through physical contact or tunnelling effect, which allows current to flow throughout the material and endows the composite with its electrical conductivity) of the composite material and consequently diminishing the sensitivity of the sensing material. The filler concentration, curing, and MT are crucial for improving the sensor sensitivity performance.

According to the ANOVA regression coefficient of BBD, the response relationship between the sensitivity, filler content, and moulding temperature can be expressed as follows:

$$\begin{gathered} y_2=3.05940-0.735600x_1-2.94273x_2-3.33888x_3-0.004671x_4\dots \\ +0.127500x_1{x}_2+0.820000x_1{x}_3+0.000713x_1{x}_4\dots \\ +1.69750x_2{x}_3+0.003813x_2{x}_4-0.006500x_3{x}_4\dots \\ -0.148167x_1^2+0.753083x_2^2+1.05058x_3^2+0.000071x_4^2 \end{gathered}$$

(44)

where $y_2$ denotes the sensor sensitivity performance response value; $x_1$, $x_2$, and $x_3$ denote the content of Fe₃O₄, CNT, and MLG, respectively; and $x_4$ is the moulding temperature.

RF

Table 13. R² values using various hyperparameters and search methodologies in forecasting sensitivity utilising RF.

Search Techniques	Number of Estimators	Maximum Depth	R2 Train	R2 Test
GridSearchCV	200	100	1.000	0.8035
	150	150	1.000	0.8018
	100	100	0.996	0.7887
RandomizedSearchCV	55	65	1.000	0.7985
	20	259	0.999	0.8046
	120	100	1.000	0.7969

presents the outcomes of the sensitivity predictions utilising the RF model. The model’s performance using the GridSearchCV approach diminishes progressively with a reduction in the number of decision trees. This is evident when comparing the three combinations with varying numbers of decision trees (200, 150, and 100), as the R² of the test set progressively declines from 0.8035 to 0.8018 to 0.7887. Conversely, altering the maximum depth does not yield a notable enhancement in performance. The parameter combinations exhibiting an identical maximum depth value of 100 yield comparable outcomes in the test set, with the R² approximately at 0.8. In the RandomizedSearchCV approach, varying the number of decision trees does not substantially enhance the model performance. The fluctuation in R² on the test set is minimal, spanning from 0.7969 to 0.8046. Furthermore, there was minimal variation in the model performance with the increase in the maximum depth. The configuration with a maximum depth of 259 and the fewest decision trees exhibited the highest R² (0.8046) on the test set. However, the configuration with a reduced maximum depth displayed slightly lower values. For all of the examined parameter combinations, the R² of the training set approaches 1. The research indicates that the model exhibits poor generalisation, likely because of overfitting; thus, further data samples may be necessary to enhance the model’s predictive accuracy.

LSTM

Figure 16a illustrates that the R² value for the training set remains relatively stable with an increased number of hidden layers. Conversely, in the test data, the R² value reaches its zenith at three hidden layers before declining with further increases in the hidden layers, indicating that three hidden layers adequately encapsulate the data’s complexity while mitigating the risk of overfitting. Consequently, the number of hidden layers was established at three, and the variation in R² concerning the amount of neurones in the hidden layers was then examined. Figure 16b illustrates that in the training data, R² remains rather stable. However, in the test data, the peak value is observed at 60 neurones per layer, yielding R² values of 0.9968 for the training data and 0.844 for the test data. Increasing the number of neurones leads to a decline in the model performance due to overfitting on the test data.

HKOA-LSTM

As illustrated in Figure 17a, akin to the hysteresis prediction outcomes, the HKOA-LSTM model utilised for forecasting the sensitivity of the flexible sensing materials demonstrates commendable performance on the training set. In the test set, the model’s R² score reaches its peak with three hidden layers and declines at the addition of a fourth layer. Consequently, increasing the number of hidden layers does not enhance the network’s generalisation capacity and may result in data overfitting. Figure 17b illustrates the impact of the neurone count per layer on model performance, with the number of hidden layers fixed at three. In the training dataset, it was seen that R² continually achieved an exceptional level as the number of neurones rose. For the test dataset, the R² value diminishes progressively from 20 to 40, increasing from 40 to 70, reaching a maximum of 70, with R² values of 1 and 0.9803 for the training and test sets, respectively.

The research indicates that the quest for high-performance sensing materials will inevitably alter their mechanical and electrical properties. The concentration of conductive particles and the MT of the sensor material influence the sensitivity and hysteresis performance, suggesting that optimising sensitivity throughout a broad pressure range may compromise the hysteresis performance. Consequently, an appropriate filler ratio and MT are necessary to equilibrate the sensitivity and hysteresis characteristics.

5. Conclusions

In this study, a composite sensing material with low hysteresis and high sensitivity was developed, which was fabricated by mechanically blending 0.665 g magnetic nanoparticles Fe₃O₄, 1.098 g CNTs, and 0.99 g MLGs as fillers, and 45 g RTV silicone rubber as the substrate. The sensing material demonstrates minimal hysteresis (3.254%) and elevated average sensitivity (0.0462 kPa⁻¹) across an extensive pressure range (0–30 kPa), in addition to remarkably low response and recovery durations (0.21/0.32 s) and outstanding repeatability. The derived mathematical model of the hysteresis and sensitivity of the composite sensing material identified the filler concentration and MT as the primary parameters influencing these features. The efficacy of various regression models (RSM, RF, LSTM, and HKOA-LSTM) utilising machine learning techniques to forecast these two attributes was assessed, and the influence of diverse parameters on the model accuracy was examined. The optimal predictive model for the hysteresis performance of sensing materials is HKOA-LSTM, with three hidden layers with 80 neurones per layer, yielding R² values of 1 for the training set and 0.981 for the test set. The HKOA-LSTM model precisely forecasts the sensitivity performance of the sensing materials, with R² values of 1 for the training set and 0.9803 for the test set. Ultimately, the MORIME algorithm was employed to produce the Pareto frontier solution set for the predictive model, and the optimal solutions derived were juxtaposed with the experimental test values, revealing minimal discrepancies, with a hysteresis error of 0.765% and a sensitivity error of 0.434%. This outcome further substantiates the prospective utility of the optimisation methodology presented in this study for sensor applications. The sensing materials prepared in this study can be applied in the fields of electronic skin, medical monitoring, and intelligent robotics due to their excellent hysteresis, sensitivity, response speed, and repeatability.

This paper’s results validate the machine learning model’s efficacy in precisely estimating and predicting the electrical characteristics of composite sensing materials. This method can significantly diminish laboratory scientific tasks, thereby conserving time and expenses related to material procurement and repetitive physical testing. It offers innovative concepts for sensor performance optimisation studies and is crucial for facilitating the large-scale industrial production of flexible pressure sensors.