Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics

Abstract

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.

1. Introduction

Machine learning has experienced rapid growth over the past few decades, bringing forth fresh opportunities across multiple application domains. These new opportunities have a significant impact on the fields of science and engineering and provide chances for paradigm shifts. Machine learning has introduced novel possibilities for modeling, predicting, and solving inverse problems related to fluid mechanics research [1].

Deep learning, as one of the most crucial branches of machine learning, has achieved remarkable success across various application domains. Particularly, deep learning has demonstrated significant accomplishments in areas such as computer vision, natural language processing, and speech recognition [2,3,4]. Deep learning algorithms can handle large-scale intricate data, extract advanced features, and attain exceptional performance in various tasks [5], such as image classification, object detection, semantic segmentation, and machine translation [6,7,8,9]. Common DNN (Deep Neural Network) models encompass a variety of architectures, including the Multilayer Perceptron (MLP) [10], Convolutional Neural Network (CNN) [11], and Recurrent Neural Network (RNN) [12] models.

Over the past several decades, CFD has undergone significant development in numerical simulations of both incompressible and compressible flows. Great progress has been achieved in CFD through the finite difference method [13], finite volume method [14], finite element method [15], and spectral method [16]. Significant advancements have been made with CFD in the domains of microscale and mesoscale methods. For instance, molecular dynamics [17], cellular automata [18], the lattice Boltzmann methods (LBM) [19], and others have contributed to these developments. As a mesoscopic CFD method, LBM is based on the evolution of statistical distributions on the lattice. Considerable success has been achieved with LBM in simulating fluid flow and associated transport phenomena [20]. In particular, LBM is able to handle a wide variety of boundary conditions and complex geometries with a simple program structure and a high degree of parallelism [21]. The most commonly used model in LBM is the single-relaxation-time LBM (SRT-LBM). Despite some limitations of the SRT-LBM, such as the challenges of flow simulation at high Reynolds numbers [21], it has become one of the most popular forms of lattice Boltzmann equations due to its simplicity [22].

Recently, numerous cases of combining deep learning with CFD numerical solution models have emerged. For example, the fusion of LBM with LSTM and ResNet models to accelerate the traditional numerical solving time [22], solving inverse problems using PINNs combined with Navier–Stokes equations [23,24,25], and tackling forward and inverse problems in fluid mechanics through solving the Boltzmann–BGK equation with PINNs [26]. Han et al. [27] employed deep convolutional neural networks to solve the Boltzmann–BGK equation, and their model was suitable for multiscale flows with Knudsen numbers ranging from 10⁻³ to 10. Da et al. [28] and utilized machine learning in combination with the LBM method to enhance simulations of flow through porous media.

Traditional numerical methods in CFD rely heavily on physical equations and assumptions for modeling, and require the application of complex numerical methods to solve a problem. The introduction of machine learning techniques informed by large-scale data-driven learning patterns and correlations has enabled systems to automatically discover and understand intricate patterns and behaviors. Through machine learning, researchers can utilize large amounts of fluid flow data to explore underlying regularities and structures and extract valuable insights. However, the computational cost is still quite high when solving the inverse problem. Different from the forward problem with a well-defined boundary and initial conditions, the inverse problem refers to scenarios with uncertain initial and boundary conditions, in which only limited passive measurements and observations are available [26]. In addition, the direct numerical simulation of turbulent systems is extremely difficult due to the need to deal with complex spatial scales and multiple physical constraints. This challenge is particularly pronounced in complex flows involving phase transitions or chemical reactions, such as thermal convection [29], as well as in various scenarios like shale gas flow in porous media [30,31], vacuum technology [32], and microfluidics [33]. In some cases, solving partial differential equations (PDEs) with million-scale grids introduces new challenges in terms of computational resources and algorithmic robustness.

Deep learning algorithms have risen as an alternative to conventional CFD methods, especially when combined with sparse data [34,35,36,37]. It is important to note that, when deep learning methods are used to solve PDEs, the solution model is consistent with solving inverse problems with fluid mechanics. Machine learning offers innovative solutions to inverse problems with fluid mechanics. Traditional approaches to solving such problems usually involve iterative optimization or statistical inference techniques. However, the training process of physics-driven neural networks aligns with the essence of solving inverse problems. By training on limited observed data, these networks can learn and deduce concealed information within fluid systems. This approach assists scientists and engineers in extracting more insights from restricted observational data, thus effectively addressing complex fluid flow issues [34]. Therefore, the advancement of machine learning techniques not only introduces novel avenues for modeling and prediction in the field of fluid mechanics but also furnishes robust tools and methods for solving inverse problems in fluid mechanics.

This paper focuses on the progress of combining PINNs with the LBM. Traditional numerical methods require discretization of the partial differential operators, whereas PINN uses back-propagation auto-differentiation techniques to compute all the necessary operators in given PDEs [26]. The combination of PINN and the LBM implies a significant reduction in computational cost by eliminating the need to generate a mesh when solving PDEs. In particular, PINNs can infer unknown parameters within PDEs and generate final solutions based on partial solutions derived from them.

PINNs are currently capable of simulating laminar and turbulent channel flows [35]. PINNs are also employed to learn the impact of density gradients on velocity, pressure, and density fields within high-speed flows, based on partial observations of the density gradient [36]. The utilization of PINNs involves inferring the velocity and pressure, as well as the temperature or solute concentration, from spatiotemporal visualizations of passive scalars and given measurements [37].

The LBM is regarded as a mesoscopic modeling approach, in contrast to the macroscopic approach embodied by the Navier–Stokes equations. The LBM operates on a lattice grid, where fluid behavior is simulated through discrete particle interactions, making it inherently mesoscopic. In contrast, the Navier–Stokes equations are macroscopic, treating fluid as a continuous medium. Furthermore, the LBM inherently captures non-equilibrium effects at the mesoscale by tracking particle distributions and collisions. The LBM has garnered substantial acclaim for its efficacy in modeling fluid dynamics and related transport phenomena. The porous media flow refers to the phenomenon of fluid movement through solid particles or porous materials. This type of flow typically involves the passage of liquids or gases through porous media with complex structures, such as soil, rock, filter materials, or sponges. The porous media flow is of significant importance in areas such as groundwater flow, petroleum extraction, environmental engineering, and filtration, as it affects processes such as mass transport, heat conduction, and mass transfer. Wang et al. employed an innovative approach by integrating machine learning with the LBM, referred to as the ML-LBM, to prognosticate the fluid flow within porous media [38]. This pioneering work amalgamated Convolutional Neural Networks (CNN) with LBM to furnish predictive capabilities for both two-dimensional (2D) and three-dimensional (3D) flow patterns. Yin et al. conducted a discerning investigation to elucidate the primary transport mechanisms governing fluid behavior within nanoporous materials, utilizing LBM as a pivotal tool in their explorations [39].

The SRT-LBM is a commonly used method in the mesoscopic LBM. The SRT-LBM has advantages in dealing with complex geometries and multi-scale flow challenges. By embedding the physical equations and boundary conditions within neural networks, PINNs can effectively learn the physical behavior of a system and make predictions at points where conditions are unknown. The combination of PINNs with macroscopic physical methods represented by the Navier–Stokes equations has excellent results [25]. There is still potential for further combinations beyond the macroscopic physical methods, such as the mesoscopic physical methods described by the SRT-LBM.

The research on using PINNs to solve inverse problems in fluid mechanics still has room for improvement. PINNs do not have the ability to quickly and flexibly apply cases of different dimensions when combined with physical methods. Moreover, the work on simulating flows is not comprehensive, especially for classical fluid mechanics. Therefore, the primary objective of this paper is to combine the SRT-LBM with PINNs and to be able to enhance the capability of PINNs in modeling incompressible steady and unsteady flows for inverse problems in fluid mechanics. In this paper, the PINN-SRT-LBM model is proposed to combine the SRT-LBM with PINNs. The novel introduction of the discrete velocity model makes the PINN-SRT-LBM model flexible enough to solve inverse problems in different dimensions.

The rest of this paper is organized as follows: In Section 2, we propose the PINN-SRT-LBM model and divide it into two sub-models, the PINN-SRT-LBM-I and the PINN-SRT-LBM-II, based on the physics-informed part. In Section 3, comparative experiments and performance testing are set up for the PINN-SRT-LBM. The last section draws conclusions.

2. Materials and Methods

2.1. Single-Relaxation-Time Lattice Boltzmann Method (SRT-LBM)

The SRT-LBM is one of the most widely used LBE models based on the Bhatnagar–Gross–Krook (BGK) collision operator [40]. The particle velocity distribution function in the SRT-LBM model is discretized in multiple velocity directions. Constructing a single-relaxation-time (SRT) model hinges on the appropriate selection of equilibrium distribution functions. The specific form of equilibrium distribution functions is contingent upon the construction of the chosen discrete velocity model (DVM). The symmetry of the discrete velocities determines whether the corresponding lattice Boltzmann model can accurately recover the macroscopic equations to be solved. Hence, the selection of the DVM constitutes a pivotal aspect in the construction of the SRT-LBM. The DnQm model (where n represents the spatial dimension and m indicates the number of discrete velocities) proposed in reference [41] serves as the fundamental model for the SRT-LBM. In the case of n = 1, the commonly used DVM is the D1Q5 model (shown in Figure 1a), and for n = 2, the prevalent choice for the DVM is the D2Q9 model (shown in Figure 1b).

The governing equations for the SRT-LBM based on the DnQm can be expressed as follows:

$$\begin{array}{c}\frac{\partial f}{\partial t}+\mathit{\xi }\times \nabla f=-\frac{1}{\tau }\left(f-f^{eq}\right)\\ f^{neq}=\left(f-f^{eq}\right),\end{array}$$

(1)

where f = f(ξ,x,t) represents the particle distribution function, t is time, x is spatial position, ξ is particle velocity, τ is the relaxation time, f^neq stands for the non-equilibrium part of the particle distribution function, and f^eq denotes the equilibrium distribution function. The specific formulas are as follows:

$$f^{eq}=\frac{\rho }{{(2\pi RT)}^{\frac{D}{2}}}\exp \left(-\frac{{\left|\mathit{\xi }-u\right|}^2}{2RT}\right),$$

(2)

where R represents the gas constant, T stands for temperature, D denotes the spatial dimension, and ρ signifies the density associated with fluid pressure p, while ρ is given by ρ = p/RT, and u represents the fluid velocity.

Since the particle velocity ξ is continuous, the DnQm discrete velocity Boltzmann model is used in this paper for computational convenience. Consequently, Equation (2) is transformed into the following discrete form:

$$f_a^{eq}=\rho {\omega }_a\left[1+\frac{{\mathit{e}}_{\mathit{a}}\times \mathit{u}}{c_s^2}+\frac{{({\mathit{e}}_{\mathit{a}}\times \mathit{u})}^2}{2c_s^4}-\frac{{\mathit{u}}^2}{2c_s^2}\right],$$

(3)

where c_s = $\sqrt{\mathit{RT}}$ represents the lattice speed of sound, ω_a stands for the lattice weight coefficient, and u represents the macroscopic velocity. When using a one-dimensional discrete velocity model, u has only one velocity component, u. When using a two-dimensional discrete velocity model, u has two velocity components, u and v. This applies to all of the vector us.

In the D1Q5 discrete velocity Boltzmann model, the discrete velocities e_a are shown below:

$${\mathit{e}}_{\mathit{a}}=\mathit{e}[0,1,-1,2,-2],{\omega }_a=\{\begin{array}{ll}\frac{1}{2},\hfill & {\mathit{e}}_{\mathit{a}}=0\hfill \\ \frac{1}{6},\hfill & {\mathit{e}}_{\mathit{a}}=1,-1\hfill \\ \frac{1}{12},\hfill & {\mathit{e}}_{\mathit{a}}=2,-2\hfill \end{array}$$

(4)

where e_a denotes the direction of velocity discretization, and ω_a stands for the weight associated with the lattice direction, a = 0, 1, 2, 3, 4.

In the D2Q9 discrete velocity model, the discrete velocities e_a satisfy the following:

$${\mathit{e}}_{\mathit{a}}=\{\begin{array}{ll}(0,0)\hfill & a=0\hfill \\ c(\cos [(a-1)\frac{\pi }{2}],\sin [(a-1)\frac{\pi }{2}])\hfill & a=1,2,3,4\hfill \\ \sqrt{2}c(\cos [(a-1)\frac{\pi }{4}],\sin [(2a-1)\frac{\pi }{4}])\hfill & a=5,6,7,8\hfill \end{array}$$

(5)

where e_a represents the direction of velocity discretization, c = $\delta x/\delta t$, and δx and δt are grid spacing and time step, respectively. The weight coefficient ω_a follows the condition:

$${\omega }_0=\frac{4}{9},{\omega }_1={\omega }_2={\omega }_3={\omega }_4=\frac{1}{9},{\omega }_5={\omega }_6={\omega }_7={\omega }_8=\frac{1}{36}.$$

(6)

In order to recover macroscopic physical quantities from the particle distribution functions in the mesoscale equation, the equilibrium distribution function f_a^eq needs to satisfy the following moment equations:

$$\begin{array}{c}{\displaystyle \sum _a{f}_a^{eq}}=\rho \\ {\displaystyle \sum _a{f}_a^{eq}}{\mathit{e}}_{\mathit{a}}=\rho \mathit{u}.\end{array}$$

(7)

where the macroscopic pressure is directly obtained from the equation of state as p = ρc_s² and c_s is the lattice speed of sound, typically set to $1/\sqrt{3}$.

2.2. Network Structure

In this part, the neural network architecture of combining SRT-LBM with PINNs is presented. To supplement the practical application of PINNs in the context of combining mesoscopic physical models, this paper proposes two variants of the PINN-SRT-LBM model, namely, PINN-SRT-LBM-I and PINN-SRT-LBM-II, based on distinct physical driving mechanisms from a mesoscopic physics perspective (refer to Figure 2 and Figure 3).

As shown in Figure 2, the PINN-SRT-LBM-I model consists of a neural network architecture and the physics-informed function of the SRT-LBM. The neural network structure of the PINN-SRT-LBM-I model comprises network inputs x and t, two deep neural networks labeled as DNN-I and DNN-II, along with the macroscopic quantities output by DNN-I and the non-equilibrium distribution function f_a^neq output by DNN-II. DNN-I and DNN-II are deep neural networks with adjustable parameters (layers and neurons). In Figure 2, k denotes the deep neural network depth. The input x represents location information and t represents time information. The network components DNN-I and DNN-II consist of deep neural networks with adjustable parameters. DNN-I and DNN-II are the fundamental neural networks that make up the PINN-SRT-LBM. The activation function of the neurons, denoted as σ, is configured as the hyperbolic tangent function (tanh) within the model. DNN-I is employed for the prediction of macroscopic quantities, while DNN-II is utilized for approximating the non-equilibrium distribution function. During training, the model’s performance can be adjusted by configuring the network’s depth and width. The output section encompasses the macroscopic quantities u and ρ, predicted by DNN-I, as well as the approximated non-equilibrium distribution function f_a^neq from DNN-II. Velocities in macro quantities are represented as a vector, which is converted to scalar form in the actual prediction, depending on the dimension to which the model applies.

The physics-informed function of the PINN-SRT-LBM-I (the SRT-LBM component in Figure 2) is initially derived using the DVM to obtain the distribution function. Subsequently, employing the automatic differentiation capability (Autograd) provided by PyTorch, the PDEs specified in the SRT-LBM formulation are computed to yield residual values. These residuals, in combination with the macroscopic quantities, formulate the Mean-Square Loss. The iterative process continues until the maximum iteration count N is reached. The definition of the residuals is obtained from Equation (1).

The construction of the PINN-SRT-LBM-II model is depicted in Figure 3. The neural network structure of this model is the same as the PINN-SRT-LBM-I model. However, in the neural network output section, DNN-I approximates the equilibrium distribution function f_a^eq, while DNN-II approximates the non-equilibrium distribution function f_a^neq. The physics-informed function of the PINN-SRT-LBM-II model employs the DVM. It transforms the equilibrium distribution function f_a^eq and the non-equilibrium distribution function f_a^neq, as defined in Equation (6), into macroscopic quantities u and ρ. These quantities are used to compute residuals through automatic differentiation. This implementation process is contrary to that of the PINN-SRT-LBM-I model. The primary distinction between the PINN-SRT-LBM-I and PINN-SRT-LBM-II models lies in the neural network outputs and the physics-informed function. Simply put, the difference between the PINN-SRT-LBM-I model and the PINN-SRT-LBM-II model is evident in their different network outputs. The former yields macroscopic quantities, while the latter provides microscopic quantities.

This difference in the PINN-SRT-LBM model stems from the variation in input values when applying physical constraints using the DVB model. Notably, the structure of the deep neural networks’ (DNN-I and DNN-II) outputs should be adjusted based on the chosen DVM. For instance, if a D2Q9 DVM is chosen, then the equilibrium distribution function and non-equilibrium distribution function (f ^eq and f ^neq) should correspond to the nine discrete velocity directions.

In both PINN-SRT-LBM-I and PINN-SRT-LBM-II, these distribution functions follow the relationship f_a (x,t) = f_a^eq + f_a^neq.

During the process of network training, challenges such as vanishing gradients (where gradients become extremely close to 0) and exploding gradients (where gradients become excessively large) can arise. These issues can lead to ineffective or counterproductive gradients during backpropagation. Consequently, in this work, the initialization method chosen for network parameters is Xavier initialization [42]. This technique is employed to mitigate the impact of gradient-related problems and contribute to the stability of the training process.

Xavier initialization demonstrates exceptional performance when the activation function is set to tanh. Xavier initialization significantly impacts the convergence speed and final performance of neural networks, potentially aiding in learning deeper levels of abstract features. Additionally, the range of weights is effectively constrained by Xavier initialization, thereby preserving the consistency of input–output data distribution variances. This special initialization helps prevent overfitting and ensures a high degree of model stability.

The formulas for Xavier initialization in both uniform and Gaussian distributions are as follows:

$$W~U[-\sqrt{\frac{6}{n_i+n_{i+1}}},\sqrt{\frac{6}{n_i+n_{i+1}}}]$$

(8)

$$W~U[0,\frac{2}{n_i+n_{i+1}}],$$

(9)

where W represents the initialized weights of the neural network and n_i signifies the size of the ith layer in the network. These initialization formulas contribute to maintaining a balanced distribution of weights, promoting better training and generalization capabilities of the neural network.

2.3. Loss Function and Optimization Method

The learning process in neural networks consists in adjusting all the biases and weights of the network in order to reduce the value of a well-chosen loss function. For the cases in this paper, it is common to choose the mean squared error.

When conducting the simulation of the inverse problem in fluid mechanics using the PINN-SRT-LBM-I and PINN-SRT-LBM-II models proposed in this paper, the physical constraints of SRT-LBM should be integrated into the training by minimizing the following loss function:

$$\begin{array}{l}L={\beta }_1\cdot L_{SRT}+{\beta }_2\cdot L_M\\ L_{SRT}=\frac{1}{N_{train}}{\displaystyle \sum _{a=0}^{Q-1}{\displaystyle \sum _{n=1}^{N_{train}}{\left|R_a({\mathit{x}}_n,t_n)-0\right|}^2}},\\ L_M=\frac{1}{N_{train}}{\displaystyle \sum _{n=1}^{N_{train}}{\left|\Phi ({\mathit{x}}_n,t_n)-{\Phi }^{*}({\mathit{x}}_n,t_n)\right|}^2},\end{array}$$

(10)

Within the loss function, L represents the sum of the loss stemming from the residuals of the SRT-LBM and the loss concerning the macroscopic quantities of the training points. L_SRT represents the residual value of SRT-LBM. Q represents the value of m in the DnQm discrete velocity model. L_M represents the loss function based on the macroscopic quantities (i.e., ρ/u/v). N_train pertains to the number of training points within the computational domain, and n denotes the number of training points. Φ signifies the macroscopic quantities (i.e., ρ/u/v). Importantly, the loss function in the introduced PINN-SRT-LBM-I and PINN-SRT-LBM-II models adheres to an identical configuration.

In the loss function (Equation (10)), this work introduces a set of weighting coefficients (β₁, β₂) to address the issue of disparate error magnitudes among different terms. These weighting coefficients ensure that distinct components of the loss function hold equal importance during the optimization process.

R_a(x,t) represents the residual at moment t of the training point at position x corresponding to the Boltzmann–BGK equation, which can be computed according to Equation (1). We construct the residual calculation formula as follows (i.e., D2Q9):

$$R_a=\frac{\partial f_a}{\partial t}+{\mathit{e}}_{\mathit{a}}\cdot \nabla f_a+\frac{1}{\tau }(f_a-f_a^{eq}),\hspace{1em}\hspace{1em}a=0,1,2,\dots ,8$$

(11)

During the training of the neural network models, two distinct neural network optimizers are employed: Adam and L-BFGS-B. These two optimizers play a crucial role in the model training process, aiding in minimizing the loss function and updating the model’s parameters.

In the training phase, a transfer learning approach using the Adam and L-BFGS-B optimizer combination was adopted to accelerate model training. Initially, during the early stages of training, the Adam optimizer is utilized with a dynamically adjusted learning rate. After the loss value is less than the pre-defined value, the optimization transitions to the L-BFGS-B optimizer and iterates to the set maximum number of iterations. This hybrid optimization strategy aims to optimize the neural network model more efficiently by leveraging the strengths of both optimizers.

2.4. Dataset Construction

In order to evaluate the model performance, the proposed PINN-SRT-LBM-I and PINN-SRT-LBM-II models were tested through simulation experiments involving 1D and 2D inverse problems in fluid mechanics. For the 1D case, the Sod shock tube problem was selected while, for the 2D cases, lid-driven cavity flow and flow around circular cylinder were selected. This section will introduce the construction of the datasets.

Initially, for the 1D Sod shock tube case, precise solution data were obtained from Riemann solvers [43]. In the case of the 2D lid-driven cavity flow, we adapted the numerical solution from the example code provided in reference [44] to create an appropriate dataset format. As for the 2D flow around circular cylinder dataset, an open-source dataset from reference [25] was utilized.

In the data processing phase, data manipulation libraries and tools were utilized for tasks such as data cleansing, preprocessing, and feature extraction. The data underwent normalization and smoothing processes to better suit subsequent data analysis and modeling tasks. It is worth noting that, in this work, the data from the numerical solution are used as the true solution for the PINN-SRT-LBM-I and PINN-SRT-LBM-II in solving the inverse problem.

Ultimately, the processed data were saved in the standardized format of Matlab data storage. The dataset encompasses crucial physical variables along with accompanying annotation information to facilitate research and analysis across different problem domains. As a reference, a schematic illustration of the dataset structure, exemplified by lid-driven cavity flow, is presented in Figure 4. This includes representative time slices of density distribution ρ(t, x, y), as well as velocity components u(t, x, y) and v(t, x, y).

Detailed information about the datasets used in the experiments is presented in

Table 1. Dataset structure.

Dataset	Xexact	ρexact	Uexact	T
Sod shock tube	Ns × D1	Ns × Ts	Ns × D1 × Ts	Ts
Lid-driven cavity flow	NL × D2	NL × TL	NL × D2 × TL	TL
Flow around circular cylinder	Ncy × D2	Ncy × Tcy	Ncy × D2 × Tcy	Tcy

, where X_exact represents the coordinate values of the points selected within the computational domain. The ρ_exact represents the recorded density of training points at different time steps, and U_exact corresponds to the velocity of points at each time step. D_i, where I = 1, 2, denotes the dimensionality, either 1D or 2D. N_s, N_L, and N_cy, respectively, indicate the number of training points in the Sod shock tube, lid-driven cavity flow, and flow around circular cylinder datasets. T_s, T_L, and T_cy correspond to the number of time steps in the Sod shock tube, lid-driven cavity flow, and flow around circular cylinder datasets. To emphasize the model’s ability to learn from scattered and sparse training data, this paper opted for N_s = 1000, N_L = 4225, and N_cy = 5000. The number of time slices was set as follows: T_s = 100, T_L = 100, and T_cy = 200. Notably, when employing our proposed model to simulate fluid mechanics inverse problems, only 1% of the available data points are utilized as training points.

3. Results

This section presents a comprehensive experimental validation and analysis of the PINN-SRT-LBM-I and PINN-SRT-LBM-II using classical CFD cases. In order to ensure accuracy, the experiments encompass both 1D and 2D cases, specifically addressing the Sod shock tube case in the 1D scenario, and the lid-driven cavity flow and flow around circular cylinder cases within the 2D scenario. Since the goal of the cases is to simulate the inverse problem using neural networks, the initial and boundary conditions do not need to be given.

The computational resources employed in this paper include NVIDIA Titan X based on the Pascal architecture. The card used in this research offers a floating-point computational capacity of 11 TFLOPS, a memory capacity of 12288MB, 3584 CUDA cores, and a clock frequency of 1.53 GHz.

3.1. Sod Shock Tube

The shock tube is a type of experimental apparatus that serves as a significant means for studying nonlinear mechanics. The Sod shock tube problem is a commonly employed test case in CFD, smoothed particle hydrodynamics (SPH), and similar methods. It serves to assess the efficacy of specific computational approaches and exposes potential shortcomings within methods, such as numerical schemes. The shock tube for the Sod shock tube problem is divided into two regions, left and right, separated by a thin membrane. When the membrane ruptures (T = 0), a shock wave propagates from the left to the right, while an expansion wave moves from the right to the left [45].

In this part, this paper employed three methods: the PINN-SRT-LBM-I, the PINN-SRT-LBM-II, and DNNs, to simulate the density field, pressure field, and velocity field within the computational region of the Sod shock tube at T = 100.

For the Sod shock tube problem, the D1Q5 DVM was utilized. The model configuration is shown in Figure 1a, and the model parameters are established according to Equation (4).

The initial values for the exact solution are set as follows:

$$\{\begin{array}{ll}({\rho }_L,u_L,p_L)=(0.125,0.0,0.1)\hfill & 0<x<0.5\hfill \\ ({\rho }_R,u_R,p_R)=(1.0,0,1.0)\hfill & 0.5<x<1.0.\hfill \end{array}$$

(12)

In the experiments, the density and length are, respectively, set as ρ = 1 and L₀ = 1. The initial conditions for the simulation use the same initial conditions as the exact solution. The grid size is set as Δx = 1/1000, the time step length is Δt = 0.001, the total duration is t = 0.1, and the number of time slices is T = 100. The position of the shock is chosen at x = 0.5.

To simulate the density, ρ, pressure, p, and velocity, u, we divide the entire simulation duration of t = 0.1 into T time slices and extract 1000 training points. This constitutes 1% of the total available data. These points are employed as internal observations for solving the inverse problem and are used for training, while the remaining data are reserved for validation.

For the DNN-I and DNN-II in both the PINN-SRT-LBM-I and PINN-SRT-LBM-II models, the neural network (DNN-I and DNN-II) configuration consists of eight hidden layers with 20 neurons each, utilizing the tanh activation function. The DNNs employed for comparison are similarly structured with eight hidden layers and 20 neurons per layer, utilizing the tanh activation function. The weight coefficients in the loss function are set as (β₁, β₂) = (1, 1). The max iteration is set to 80,000, and the optimizer follows the Adam + L-BFGS-B combination.

Figure 5 shows a comparison of the training loss curves for the Sod shock tube inverse problem simulation using three methods: the PINN-SRT-LBM-I, the PINN-SRT-LBM-II, and DNNs. The depicted oscillation amplitude and convergence range of the training loss curves in the graph indicate a significant advantage of the PINN-SRT-LBM-I.

To further validate the superiority of the PINN-SRT-LBM-I model,

Table 2. The relative L₂ errors for PINN-SRT-LBM-I, PINN-SRT-LBM-II, and DNNs.

Model	erru (%)	errp (%)	errρ (%)
PINN-SRT-LBM-I	6.47%	1.64%	1.85%
PINN-SRT-LBM-II	15.87%	10.56%	14.87%
DNNs	18.87%	5.66%	5.75%

presents the relative L₂ errors of the simulation results obtained using the three methods with respect to the reference solution. Here, err_u, err_p, and err_ρ represent the relative L₂ errors for the velocity, pressure, and density, respectively.

The data in Table 2 reveal that the PINN-SRT-LBM-I model exhibits higher accuracy in solving the Sod shock tube problem compared to DNNs and PINN-SRT-LBM-II. This advantage translates to more precise simulation results within the predicted region and enhanced stability.

The definition of the relative L₂ error is given by:

$$er{r}_{L_2}=\frac{1}{N}{\displaystyle \sum _{i=1}^N|y_i-y_i*{|}^2},$$

(13)

where y_i represents the predicted value, y_i* represents the exact value, and N is the number of prediction points.

Furthermore, Figure 6 presents a schematic curve depicting the variation in the loss function values during training for the PINN-SRT-LBM-I model. The curves include Loss SRT, representing the loss function of the residual values from the SRT-LBM’s PDEs; Loss Macroscopic, depicting the loss function of the macroscopic quantities at training points; and Loss Total, the summation of the preceding two. By comparing the training loss curves over iterations, one can observe the substantial influence of the Loss SRT curve on the Loss Total curve. This validates that the proposed model in this paper enforces the physical constraints on neural network training.

In Figure 7, a comparison is presented between the predicted pressure p, velocity u and density ρ, by the PINN-SRT-LBM-I model and the exact solutions.

In order to simulate the inverse problem, this work verifies the accuracy of the model predictions by using the solution obtained from the numerical simulation as the true solution of the inverse problem. The results in Figure 7 indicate that the predicted physical quantities closely match the exact solutions, highlighting the effectiveness of the PINN-SRT-LBM-I in simulating the Sod shock tube inverse problem. However, it is worth noting that, when abrupt changes occur in the physical quantities over a short time, the predictions display oscillations. This suggests that the neural network’s ability to learn such rapid variations is limited due to the nonlinear nature of these changes. To improve this aspect, exploring networks with stronger nonlinear learning capabilities is advisable.

3.2. Lid-Driven Cavity Flow

In this section, we conduct experimental validation concerning a classic problem in 2D scenarios: the incompressible steady-state lid-driven cavity flow. The experiments are conducted in a 2D cavity domain, denoted as Ω = (0,1) × (0,1). The number of time slices is set to T = 100, a continuous and constant rightward initial velocity of u = 0.1, v = 0 is applied at the upper boundary of the cavity. Non-equilibrium extrapolation is employed for the boundary conditions. The initial density of the flow field is ρ₀ = 1, and the governing equation is represented by Equation (1). The DVM utilizes the D2Q9, as illustrated in Figure 1b.

There are four datasets of training data, each comprising 4225 randomly selected spatiotemporal training points within the computational domain, which are gathered at Re = 400, 1000, 2000, and 5000. The reference solutions for the velocity field and density field of these points are derived from reference [44]. The horizontal velocity and vertical velocity are denoted as u(T, x, y) and v(T, x, y), respectively, while the density field is represented as ρ(T, x, y). The weights for the loss function are set as (β₁, β₂) = (1,1). In this reference solution, 1% of the entire dataset will be used for training, with the remaining portion utilized for validating the predictive outcomes. Considering that the flow field of the lid-driven cavity under the specified Reynolds numbers attains a steady state, we focus solely on simulating the flow field at the final time step (T = 100).

To train the models for the experimental cases, the parameter configurations of DNN-I and DNN-II, the base components of the PINN-SRT-LBM-I and PINN-SRT-LBM-II models, are as follows: eight hidden layers, each containing 40 neurons, and a hyperbolic tangent (tanh) activation function. The parameters of the contrasting DNNs are set to eight layers with 40 neurons per layer. The maximum number of training iterations is established at 80,000, and the training process employs the Adam + L-BFGS optimizer. Notably, for the DNNs in this experiment, the SGD optimizer will be utilized in the pursuit of a potentially enhanced performance.

The results shown in Figure 8 unveil the characteristics of the training loss curves for the three methods, the PINN-SRT-LBM-I, the PINN-SRT-LBM-II, and DNNs, when simulating the lid-driven cavity flow. Notably, the loss value of the DNNs experiences a rapid descent to below 1 × 10⁻³ followed by an onset of overfitting. Meanwhile, the loss value of the PINN-SRT-LBM-II model converges within a range one order of magnitude smaller than that of the PINN-SRT-LBM-I model. This deduction is grounded in the fact that the PINN-SRT-LBM-II model directly leverages deep neural networks to approximate both equilibrium and non-equilibrium distribution functions. When tackling the 2D inverse problem of the lid-driven cavity flow, the PINN-SRT-LBM-II model is anticipated to excel in capturing intricate details of the underlying physical distribution patterns compared to both DNNs and the PINN-SRT-LBM-I model. In order to empirically validate our conjecture,

Table 3. The relative L₂ errors for PINN-SRT-LBM-I, PINN-SRT-LBM-II, and DNNs at Re = 1000 and T = 100.

Model	erru (%)	errv (%)	errρ (%)
DNNs	2.01%	1.76%	0.23%
PINN-SRT-LBM-I	0.30%	0.26%	0.05%
PINN-SRT-LBM-II	0.08%	0.06%	0.03%

presents the relative L₂ errors for the results obtained by the PINN-SRT-LBM-I, PINN-SRT-LBM-II, and DNNs when simulating at Re = 1000 and T = 100.

In Table 3, err_u, err_v, and err_ρ, respectively, denote the relative L₂ errors of u, v, and ρ. The results notably illustrate that the relative L₂ errors of the predictions generated by the PINN-SRT-LBM-II model are approximately one order of magnitude lower than those of both DNNs and the PINN-SRT-LBM-I model.

For a more comprehensive comparison of the accuracy of different models, Figure 9 displays the visualizations of the absolute errors between the reference solution and the simulated results by the PINN-SRT-LBM-I, PINN-SRT-LBM-II, and DNNs. The first column, (a), (d), and (g), presents the visualizations of the absolute errors for the PINN-SRT-LBM-I; the second column, (b), (e), and (h), displays the visualizations of the absolute errors for the PINN-SRT-LBM-II; and the third column, (c), (f), and (i), showcases the visualizations of the absolute errors for DNNs. The absolute error images reveal the observable discrepancies between the predicted and reference values.

As for the absolute error, let Y_pred(T, x, y) denote the predicted value at coordinates (x, y) for T = t, and Y_exact(T, x, y) represent the reference solution value at the same coordinates and time. The absolute error function is defined as follows:

$$E_{abs}(t,x,y)=|Y_{pred}(t,x,y)-Y_{exact}(t,x,y)|,$$

(14)

where E_abs(t, x, y) represents the absolute error function, Y_pred(t, x, y) denotes the predicted value at time T = t and position (x, y), and Y_exact(t, x, y) signifies the reference solution value at time T = t and position (x, y). Subtracting these two values and taking the absolute value yields the absolute error at point (x, y) during the time slice T = t.

From the results shown in Figure 9 and the relative L₂ error presented in Table 3, a reasonable inference can be made that DNNs lack the capability to address the inverse problem of 2D lid-driven cavity flow. Moreover, it can be observed that the PINN-SRT-LBM-I model, when employed to simulate the 2D lid-driven cavity flow, struggles to capture the characteristics of physical quantities at the boundaries. This phenomenon arises due to the significant variations in physical quantities in proximity to the boundaries. Consequently, the model encounters greater difficulty in learning the evolution patterns near these boundaries compared to other regions, leading to elevated levels of error.

It is worth noting that, in the simulations of CFD, the Reynolds number exerts a substantial influence on the results. Thus, in the experimental cases of this section, we have included investigations of the inverse problem of lid-driven cavity flow under various Reynolds numbers.

In the course of the experiments, four distinct Reynolds number values were considered: Re = 400, Re = 1000, Re = 2000, and Re = 5000. Predictions were made for the results at T = 100, which corresponds to the attainment of a steady state within the cavity. Using the PINN-SRT-LBM-II to address predictions for different Reynolds numbers, the variation in the training loss curves with respect to time is depicted in Figure 10. Observing the trend of the loss value across iterations in Figure 10, it can be concluded that the convergence range of the model’s loss value remains relatively consistent as the Reynolds number increases. This indicates that the PINN-SRT-LBM-II model exhibits stability across different Reynolds numbers and underscores its capacity for generalization.

In Figure 11, we present the visual results of the predictions made by the PINN-SRT-LBM-II model and the reference solution for Re = 5000 and T = 100.

Figure 11a–c respectively depicts the visualizations of the reference solution ρ, the predicted ρ from the PINN-SRT-LBM-II model, and the visualization of the absolute error in the predicted ρ. Figure 11d–f correspondingly represents the visualizations of the u for the reference solution, the predicted u from the model, and the visualization of the absolute error in the predicted u using the PINN-SRT-LBM-II model. Similarly, Figure 11g–i shows the same pattern for v.

In Figure 11b,e,h, elevated absolute error values in regions near the cavity boundaries and the primary vortex can be observed. Notably, the absolute errors surpass 10⁻² at the cavity vertices, with errors near the primary vortex exceeding other areas by around 10⁻³. These discrepancies are attributed to the accumulation of errors and rapid changes in the physical distribution patterns.

Significantly, the absolute error results in the predicted density, ρ, are notably better than those for u and v in the results. Consequently, we can infer that regions with intense variations in physical patterns within the computational domain pose a challenge for the model’s learning process. Moreover, the mesoscopic physics-based approach SRT-LBM imparts a distinct advantage to the model in accurately capturing the distribution patterns of ρ.

This is consistent with the results presented in Figure 11, Figure 12, Figure 13 and Figure 14, which depict visualizations of the reference solutions, model predictions, and absolute error results for Reynolds numbers Re = 400, 1000, and 2000 at T = 100.

Through the progressive comparison of Figure 11, Figure 12, Figure 13 and Figure 14 as the Reynolds number increases, a conclusion can be drawn. Given that the maximum absolute error remains below 10⁻², the PINN-SRT-LBM-II model exhibits stability in simulating 2D lid-driven cavity flow. Furthermore, a comparative analysis of the results reveals that regions with more pronounced variations in the physical distribution patterns tend to exhibit relatively higher absolute errors compared to other areas within the computational domain. This is due to the escalating nonlinearity with higher Reynolds numbers. However, the nonlinearity within the PINN-SRT-LBM-II model does not align perfectly. For instance, in Figure 11c,f,i, there are irregular oscillations and uneven distribution of absolute errors within the simulation domain. As a result, the prediction accuracy gradually decreases with higher Reynolds numbers. In order to address the above, attention should be given to countering overfitting during the model’s training process [46].

To better assess the alignment between the results of the PINN-SRT-LBM-II model and the reference solution, this paper provides further comparison in

Table 4. Comparison between the predicted center coordinates of the primary vortex by the PINN-SRT-LBM-II model and the reference results.

Re	A	B	C	D
400	(0.5563, 0.6000)	(0.5547, 0.6055)	(0.5608, 0.6078)	(0.5556, 0.6000)
1000	(0.5438, 0.5625)	(0.5313, 0.5625)	(0.5333, 0.5647)	(0.5327, 0.5652)
2000	(0.5226, 0.5482)	(0.5255, 0.5490)	(0.5250, 0.5500)	(0.5254, 0.5499)
5000	(0.5125, 0.5313)	(0.5117, 0.5352)	(0.5176, 0.5373)	(0.5137, 0.5424)

regarding the coordinates of the main vortex center point. A represents the predictions from the PINN-SRT-LBM-II, while B, C, and D correspond to the results presented in references [47,48,49,50].

The conclusion drawn from the coordinates of the main vortex center point is that there exists a strong physical consistency between the prediction results and the reference solution. This signifies that the predictive results of the PINN-SRT-LBM-II model possess certain physical attributes, indicating the capability of the model to incorporate physical constraints, which is a manifestation of the synergy between the PINN-SRT-LBM-II model and SRT-LBM principles.

The relative L₂ errors of the simulation results are shown in

Table 5. The relative L₂ errors for PINN-SRT-LBM-II at Re = 400, 1000, 2000, 5000.

Re	erru (%)	errv (%)	errρ (%)
400	0.06%	0.09%	0.05%
1000	0.08%	0.06%	0.04%
2000	0.09%	0.06%	0.04%
5000	0.13%	0.05%	0.04%

. The results indicate that, with an increase in the Reynolds number, the relative L₂ errors do not exhibit a pronounced rise. This further substantiates the physical reliability and stability of the PINN-SRT-LBM.

This work has differentiated the dimensions for which the PINN-SRT-LBM-I and PINN-SRT-LBM-II models are best-suited through comparative experiments. The PINN-SRT-LBM has successfully conducted simulations of the inverse problem of 2D lid-driven cavity flow by learning from a sparse, randomly distributed 1% of data within the computational domain. This extension from one dimension to two dimensions demonstrates the applicability of dispersed temporal and spatial data, akin to the passive scalar or limited data encountered in inverse problems, within the training of neural network models.

3.3. Flow around Circular Cylinder

In this part, we opted to employ a 2D flow around circular cylinder dataset provided by M. Raissi et al. in reference [25]. Flow around circular cylinder is a classic simulation problem in CFD. The purpose of conducting experiments in this subsection is twofold: to verify the performance of the PINN-SRT-LBM in simulating the inverse problem of incompressible unsteady flows and to conduct comparative experiments with the work of other researchers. Open-source code exists for PINNs solving the inverse problem of fluid mechanics using the Navier–Stokes equations (referred to as PINN-NS in the subsequent). The performance of the PINN-SRT-LBM in addressing the inverse problem of unsteady flows will be elaborated upon in detail within this part.

The dataset was set up with Re = 100, focusing on the wake region of the flow around circular cylinder. The dataset contained 5000 sampled training points, with a total of T = 200 time slices. For simplicity, the data collection was confined to a rectangular region downstream of the cylinder, as shown in Figure 15. The solver employed in this case was the spectral/hp element solver NekTar [50]. Since the dataset only contained velocity and pressure, the focus in this part’s experiment was solely on simulating the velocity u(t, x, y) and v(t, x, y) within the sampled region.

This case assumes that a uniform freestream velocity profile is imposed at the left boundary of the domain Ω = (−15, 25) × (−8, 8). A zero-pressure outflow condition is applied at the right boundary, 25 units downstream of the cylinder. The top and bottom boundaries of the domain adopt periodic boundary conditions. The initial density ρ = 1. The initial velocity of the fluid is u = 1, v = 0, the cylinder diameter D = 1, and the viscosity coefficient ν = 0.01. The system displays periodic steady-state features characterized by asymmetric vortex shedding patterns in the wake flow around circular cylinder, known as the Kármán vortex street [51]. The simulation is confined to the region (1, 7) × (−2, 2).

For the training settings, the parameters of DNN-I and DNN-II within the PINN-SRT-LBM-I and PINN-SRT-LBM-II were configured with eight hidden layers, each containing 40 neurons and utilizing the tanh activation function. The DNNs employed for comparison were also designed with eight hidden layers and 40 neurons per layer. The PINN-NS’s network configuration followed the default structure outlined in [25], which entails six layers with 40 neurons each, utilizing the tanh activation function. Regarding optimizer selection, the DNNs were optimized using the SGD optimizer, while the other models employed a combination of the Adam optimizer and L-BFGS optimizer. The maximum number of iterations was uniformly set to 100,000.

Figure 16 presents a schematic of the variations in the training loss curves for the three models: DNNs, the PINN-SRT-LBM-I, and the PINN-SRT-LBM-II. From the trend of the loss value shown in Figure 16, it can be concluded that the PINN-SRT-LBM-II model exhibits a more distinct convergence range when simulating the inverse problem of 2D flow around circular cylinder. This suggests that the precision of the PINN-SRT-LBM-II model is higher compared to the other models. Additionally,

Table 6. The relative L₂ error at T = 100.

Model	erru (%)	errv (%)
DNNs	13.48%	20.51%
PINN-SRT-LBM-I	2.71%	2.47%
PINN-SRT-LBM-II	0.49%	0.51%
PINN-NS	0.35%	0.72%

presents the relative L₂ errors for u and v at T = 100.

Conclusions can be drawn from the relative L₂ errors in the results from the four models presented in Table 6. Under the same training conditions, our proposed PINN-SRT-LBM-II model achieves a level of accuracy comparable to that of the PINN-NS model in [25]. Additionally, the PINN-SRT-LBM-II outperforms the PINN-SRT-LBM-I in simulating the 2D inverse problem. Consequently, for the subsequent experiments, we will employ the PINN-SRT-LBM-II model.

Flow around circular cylinder is a classic unsteady flow phenomenon. In this paper, an attempt is made to verify the learning effect of the PINN-SRT-LBM in the whole continuous time domain. The predictions using the PINN-SRT-LBM-II model were performed at T = 50, 100, and 200. Refer to Figure 17, Figure 18 and Figure 19 for the respective results.

Figure 17, Figure 18 and Figure 19 (a),(b) represent the reference solution and model predictions for u (velocity component in the x-direction), (c) and (d) depict the reference solution and model predictions for v (velocity component in the y-direction), and (e) and (f) show the visualizations of the absolute errors for u and v, respectively.

Observing the absolute errors in panels (e) and (f) of Figure 17, Figure 18 and Figure 19 leads to a reasonable inference. Due to error accumulation, areas with larger absolute errors are predominantly located in regions of significant fluid variation, aligning well with the flow patterns in the cylinder wake flow. The shedding of the Kármán vortex street in the wake exhibits the most pronounced variations, posing a challenge for neural networks to capture the underlying physical distribution patterns. This issue can potentially be addressed through alterations in training methodologies and the optimization of neural network structures. Upon sequential comparison at time snapshots T = 50, 100, and 200, a noteworthy observation emerges when focusing on the early time slice (T = 50), as shown in Figure 17. Due to the absence of a relatively stable flow configuration and the periodic shedding of the Kármán vortex street, the PINN-SRT-LBM-II model’s predictions exhibit oscillations in the distribution of absolute errors within the computational domain, owing to the abrupt variations in the underlying physical distribution patterns.

It is noteworthy that the absolute error values in the model predictions shown in Figure 17, Figure 18 and Figure 19 do not exceed magnitudes of 1 × 10⁻¹. Furthermore, the majority of the computational domain exhibits absolute errors in the model predictions that are smaller than 1 × 10⁻². From this, the conclusion can be drawn that the PINN-SRT-LBM-II model is capable of providing highly accurate prediction results across the continuous temporal and spatial domain of the unsteady 2D inverse problem.

To validate the physical reliability of the prediction results generated by the PINN-SRT-LBM-II model, Figure 20 provides streamline diagrams for T = 50, 100, and 200. In Figure 20, panels (a) and (b) depict the streamline patterns of the reference solution and model predictions at T = 50; panels (c) and (d) display the patterns at T = 100; and panels (e) and (f) illustrate the patterns at T = 200. By comparing the streamline patterns of the reference solution with those of the model predictions, it becomes apparent that the positions of the vortex shedding are nearly identical. While the dataset only captures a portion of the wake, excluding the performance near the boundary, the congruence between the vortex shedding positions and streamline patterns between the reference solution and the model predictions still underscores the physical reliability of the results obtained from the PINN-SRT-LBM-II model.

Finally, in Figure 21, we present the visualizations of the absolute errors for u and v, simulated by PINN-NS and the PINN-SRT-LBM-II. From the visualizations, a reasonable deduction can be made that, under the same training parameter settings, the accuracy of the prediction results from the PINN-SRT-LBM-II is comparable to that of PINN-NS. At the boundaries of the simulation region, the performance of the PINN-SRT-LBM-II is better than PINN-NS. However, in regions characterized by more pronounced variations in physical behavior, the predictive accuracy of the PINN-SRT-LBM-II is at a disadvantage. Overall, the accuracy of simulations conducted by the PINN-SRT-LBM-II model is approaching that of PINN-NS. This underscores the potential of mesoscopic physics methods in conjunction with PINNs, except macroscopic physics methods, and demonstrates the feasibility of employing mesoscopic physics methods combined with PINNs for simulating inverse problems in fluid mechanics.

In this part, this paper employed the PINN-SRT-LBM-II model to test its performance in simulating 2D unsteady fluid mechanics inverse problems. Moreover, for an impartial assessment of the PINN-SRT-LBM-II model’s capabilities, we compared its prediction accuracy with that achieved by existing PINNs coupled with macroscopic physics methods, particularly the Navier-Stokes equations. The test data and comparative results collectively indicate the substantial potential of the PINN-SRT-LBM-II model in inverse problem simulations. Notably, in the realm of simulating 2D inverse problems in fluid mechanics, the performance of the PINN-SRT-LBM-II model has already reached the same level as that presented in reference [24], where PINNs were combined with the Navier-Stokes equations.

4. Conclusions

To solve the inverse problem in fluid mechanics, this paper proposes the PINN-SRT-LBM by combining the SRT-LBM with PINNs. The model proposed in this work has the capability of extracting the features of the flow under the constraints of the physical laws and providing spatio-temporal simulation results. For the 1D problem, this paper proposes the PINN-SRT-LBM-I and, for the 2D problem, this paper proposes the PINN-SRT-LBM-II. The combination of the SRT-LBM with deep neural networks is achieved by reducing the Mean Square Loss which consists of the physical residual values and the existing macroscopic quantities.

The comparison tests demonstrate that the PINN-SRT-LBM-I has the advantages of high accuracy and stable prediction results in simulating the 1D inverse problem, and the PINN-SRT-LBM-II has the advantages of high physical reliability in simulating the 2D inverse problem, with the advantages of high accuracy as well. The two sub-models of the PINN-SRT-LBM can flexibly solve complex inverse problems in fluid mechanics under different scenarios.

To further test the performance of the proposed model, this paper conducts comparisons of the PINN-SRT-LBM with DNNs and PINN-NS, and the results show the scalability of the PINN-SRT-LBM. The comparison experiments in different dimensions show that the PINN-SRT-LBM is suitable for both unsteady and steady-state flow simulations, exhibiting excellent versatility.

The combination of PINNs and the LBM can be used to solve the inverse problem in fluid mechanics, and the model proposed in this paper has good scalability and extensibility in different dimensions. In particular, the PINN-SRT-LBM is applicable to 1D and 2D unsteady-flow and steady-flow inverse problems. The PINN-SRT-LBM also has the potential to solve 3D problems by replacing the DVM with D3Q19, D3Q20, D3Q27, etc. Future work will be focused on promoting the combination of a multi relaxation time LBM (MRT-LBM) and PINNs, as enabling the combination of PINNs and the LBM can solve complex fluid flow problems more accurately. The combination of PINNs and the LBM will be applied to various fields in more complex fluid simulations such as porous media flow, multiphase flow, micro flow, etc. The potential of PINNs for solving inverse problems in fluid mechanics can be further developed in future work by evaluating the neural network model or replacing the physical method.