Application of Neural Network Algorithms for Central Wavelength Determination of Fiber Optic Sensors

Featured Application

Interrogation systems of fiber-optic sensors.

Abstract

Fiber Bragg gratings are sensitive elements in fiber optic sensor networks, and this paper discusses the practicalities of using neural network algorithms to determine their central wavelengths. The problem is to determine the central wavelength of a single sensor, the parameters of which are obtained using a low-resolution spectrum analyzer. The configuration of the neural network and the algorithm for producing the training and control datasets are specified. The training results for the selected neural network configuration demonstrated that the proposed method could determine the position of the central wavelength with a resolution two and a half orders of magnitude higher than the resolution of the input data sampling. The obtained results demonstrate that the approach makes it possible to determine the FBG central wavelength shift with an error not exceeding ~0.5 pm at a spectrum analyzer resolution of 167 pm.

1. Introduction

Fiber optic sensor system design of the point and/or quasi-distributed type is impossible without solving the key problem for this type of system, which is finding the central wavelength in the reflection spectrum with the most accuracy. Thus, the shift of the central wavelength of a fiber optic sensor based on a resonant structure (a fiber Bragg grating (FBG) [1,2,3], a Fabry–Perot interferometer [4,5,6], or a ring resonator [7,8,9,10]) or other types of resonators [9,11] determines the magnitude of the physical impact. In addition, temperature [9,12,13], pressure [14,15], deformation [16,17], friction, wear, and other types of impacts [18,19] are all determined using information about the position of the sensing element’s reflection spectrum.

The primary factor limiting the estimation accuracy of the physical impact magnitude in fiber optic sensor systems is the inadequate resolution of the devices used to measure the spectral properties of such sensors. Increasing the resolution of a device’s built-in spectrum analyzer can improve its technical capabilities, but this usually comes at a cost that rises almost exponentially.

Diverse methods have been proposed to improve the precision of determining the position of the sensing elements’ spectra. Thus, the sub-pixel spectrum processing techniques [20,21,22] made it possible to increase the resolution by one or two orders of magnitude [23]. Meanwhile, after adjusting for the nonlinear spectrum distortions in charge-coupled devices depending on the signal accumulation time, the resolution can be improved by another order of magnitude but only in an extremely narrow frequency range [24]. Due to additional mathematical processing of the data, the maximum results obtained indicated that the physical resolution of the devices could be increased by nearly two orders of magnitude [23,24]. However, the achieved capabilities to determine the magnitude of the spectral shift are insufficient for some applications. The spectral shift can be determined with greater precision with the help of sensitive elements with narrow spectral characteristics. Fiber Bragg structures with discrete phase shifts, whose reflection spectra are an order of magnitude narrower than those of fiber Bragg gratings [25,26,27], and optical elements based on ring resonators, whose Fano resonance has a very narrow reflection spectrum [8,11,28], fall into this category of sensitive elements. However, high-resolution spectrum analyzers are also needed when working with sensitive elements that exhibit narrow spectral characteristics.

Microwave-photonic methods [20,29,30] are another way to improve the accuracy of measurements. These methods work by transferring the measurement process from the optical range of the electromagnetic spectrum to the microwave one. Although microwave-photonic methods are highly accurate for physical parameters determination, they do necessitate upgrading to new technologies and replacing the current hardware and software of measurement instruments.

A number of researchers have proposed the artificial intelligence algorithms to determine a central wavelength of sensors based on fiber Bragg gratings (

Table 1. Accuracy of neural network methods for FBG interrogation.

Reference	Wavelength Estimation Error	Temperature Estimation Error	Data Resolution
[31]	1.45 pm (RMSE)	0.145 °C	1 pm
[32]	2.081 pm (RMSE)	0.2081 °C	-
[33]	2 pm (2∙σ)	0.2 °C	-
[34]	2 pm (mean), 25 pm (σ)	0.2 °C	4 pm
Current work	0.411 pm (MAE), 0.533 pm (σ)	0.05 °C	167 pm

). L. Negri et al. [31] proposed to use a fully connected (FCC, fully connected cascade) neural network trained by the Neuron-by-Neuron algorithm to approximate the FBG reflection spectrum and eliminate spectrum distortions, the spectra of which were obtained with a resolution of 1 pm. They managed to achieve an error in the FBG central wavelength determination of 1.45 pm using the approximation of the FBG spectrum curve using the neural network. Y. An et al. [32] used a feed-forward neural network (BPNN) with an L-M optimization algorithm to determine the temperature using FBG. Based on the experimental data of the FBG reflectance spectra, the achieved accuracy was 0.2081 °C (root-mean-square error, RMSE), 0.8943 °C (max absolute error) or 2.081 pm (RMSE), 8.943 pm (max absolute error) on the training set. A. Zimmermann et al. [33] used a neural network (ANN) with three FBG sensors and three photodetectors to expand the temperature measurement range (from 25 °C to 250 °C). The accuracy obtained was ±0.4 °C (2σ, where σ is the standard deviation in normal distribution) or ±4 pm up to 150 °C and ±0.9 °C (2σ) or ±9 pm above 150 °C. A. Paterno et al. [34] reported on the use of a network of radial basis functions for approximating the FBG reflectance spectrum. The neural network was trained according to the algorithm described by Orr [35]. Using the approximation of a spectrum consisting of 1000 points, an accuracy of 2 pm (mean error) and 25 pm (σ) was achieved on an interval of 4 nm.

In this work, in contrast to [31,32,33,34], we proposed to use a neural network of a simpler configuration to increase the processing speed of FBG sensors in the interval Δ under study, the spectra of which were taken at a low resolution of the spectrum analyzer (~167 pm/pixel), and the aim was the accuracy of the FBG central wavelength shift determination with an error not exceeding 1 pm. The obtained accuracy corresponded to 0.04 °C (mean absolute error, MAE) and 0.05 °C (σ). Looking ahead, we include in Table 1 the data obtained from our study to compare them with those of other researchers.

As the processing power of computers has increased over the past few decades, artificial neural networks and other machine-learning techniques have become widespread in all areas of science and technology. However, while many articles (e.g., [31,36,37,38]) demonstrate results when similar approaches are used, this does not exhaust the potential of artificial neural networks.

2. Problem Statement

In our situation, we were using an I-MON-512 Interrogation unit [24]. We assumed a fiber Bragg grating (FBG) with a length L ranging from 0.5 to 3.5 mm, a period Λ varying from 531.111 to 531.472 μm, and an induced refractive index n ranging from 0.5∙10⁻⁴ to 1.5∙10⁻⁴, which was subjected to a specific physical effect (stretching/compression, temperature, etc.) causing a shift of the FBG central wavelength. With such parameters, the FBG central wavelength λ_B ranged from 1549.5 to 1550.5 nm, and the full width at half maximum of the spectrum ranged from 0.229 to 1.458 nm. The broadband radiation from a superluminescent diode was directed to the FBG via a fiber optic circulator, and the I-MON-512 spectrum analyzer received the reflection spectrum. The left and right boundaries of the controlled spectral range were λ_min = 1510 nm and λ_max = 1595 nm, respectively; the number of points in the spectrum was N = 510; and the discretization interval was Δλ = (λ_max − λ_min)/N ≈ 0.167 nm.

Without knowing the true FBG central wavelength, the data collected from the I-MON-512 cannot be used as input data for the neural network model. So instead, we used a mathematical model based on the homogeneous layers method [39] and transfer matrices [40] to obtain the FBG spectra with pre-defined central wavelengths. Initially, the FBG reflection spectrum was formed based on a mathematical model using the mathematical apparatus of transfer matrices [40] or the homogeneous layers method [39]. Then, the discretization of the modeled spectrum was performed with an interval equal to that of the spectrum analyzer. The resulting data array was essentially a discrete spectrum similar to that obtained via the interrogation device, and at the same time, the exact value of its central wavelength was known. Thus, the mathematical modeling apparatus enabled precise comparisons between the discrete spectrum and the corresponding central wavelength. These were the datasets used to train the neural network.

The position of the FBG spectrum was studied in a controlled wavelength range λ_B ± Δ/2, at Δ = 3 nm, which determined, taking into account the spectrum analyzer’s resolution, the dimension of the neural network input data M = int(Δ/Δλ) + 1. Within the scope of the problem just described, this range included 19 spectrum values, determining 19 input values of the neural network. The spectra of the same FBG for two different positions (red and blue lines) are schematically shown in Figure 1. The solid line shows the true FBG spectra (as if they were obtained with any arbitrarily small discretization interval), and the dots show the values of the spectra obtained discretely. Figure 1 uses the following designations: S^r and S^b are spectra at two positions; λ_i is the wavelength, λ_L is the wavelength of the left, and λ_L+M−1 is the right boundary of the controlled part of the spectrum; Δλ is the discretization interval; and D is the relative shift vector of the spectrum (S^b vs. S^r), which is the same for all spectrum points.

To bring the mathematical model of the spectrum as close as possible to the actual physical spectrum, the model incorporated the noise component and the minimum threshold of the light response. The amplitude of the noise component a_N was assumed to be less than 0.2% of the maximum amplitude, and the minimum value in the FBG spectrum was assumed to be less than 20% of the maximum amplitude.

To make the solution of to the problem universally applicable, let us replace the variables:

$${\tilde{\lambda }}_i=\frac{{\lambda }_i-{\lambda }_L}{{\lambda }_{L+M-1}-{\lambda }_L}, {\tilde{R}}_i=\frac{R_i-\min \left(R\right)}{\max \left(R\right)-\min \left(R\right)},$$

(1)

thus moving the input data into the range of amplitude and wavelength variation from 0 to 1. In Equation (1), the set R denotes the set of reflection coefficients R_i, i = 0.509, belonging to the wavelength range λ ∈ [1510, 1595] nm. Let us omit the tilde sign for normalized data to make future recording easier. We also assume that we are dealing with normalized values by default.

The normalization of the spectrum values by wavelength and amplitude enabled us to make the solution universal and apply it to FBG wavelength determination in any frequency range, assuming the similarity criterion was met.

An elementary dataset is a collection of M normalized discrete values of the spectrum {R_i} (i = 0, M − 1) obtained through mathematical modeling. The target value of the central wavelength λ_B corresponds to this dataset.

As a neural network, we used one of the most commonly used architectures in data approximation, linear and nonlinear regression problems—a three-layer Rosenblatt’s perceptron with variable S-A connections [41]. The three layers were sensory S (input), associative A (hidden), and response R (output). A distinguishing feature of Rosenblatt’s perceptron with sequential links is the synaptic connectivity of each neuron of the current layer with all neurons of the next layer [41]. Only associative and response layers can be trained in this configuration. The computational model used in this problem was configured from 19 elements of the sensory layer, 10 elements of the associative layer, and one element of the response layer, as shown in Figure 2.

The computational difficulty of the proposed neural network can be estimated by counting the number of multiplication and addition operations required. For example, the direct signal propagation algorithm through the neural network can be estimated as (n − 1) matrix multiplications and (n − 1) vector additions, where n is the number of model layers. Therefore, the multiplicative computational complexity of the algorithm is defined as O(w), and the arithmetic complexity is of the order O(2w + v), where w is the number of synaptic connections and v is the number of neuron shifts equal to the number of neurons of the associative and response layers. Thus, for the neural network used, the computational complexity was O(288). This value ensures near-instantaneous network response, which is especially important when running real-time sensor systems.

The problem was simplified to calculating the true FBG center wavelength with an error less than ~Δλ∙10⁻³, having only spectrum data {λ_i, R_i} (i = 0, M − 1).

3. Parameter Selection

A significant problem, which influences the neural network training speed and convergence of the neural network response to the target function, is the choice of method for optimizing the network parameters, the activation function of neurons, and the choice of a criterion for evaluating the result (the loss minimization function). Modern optimization methods in linear and nonlinear regression and approximation problems use two main methods for minimizing the loss function: the stochastic gradient descent method [42] with its varieties and the accelerated Nesterov gradient method (the pulse accumulation method) [43]. Each ensures that the neural network training procedure quickly converges to the target value (target function).

The Adam optimization method [44] was used in this paper, combining the ideas of momentum accumulation and adaptive learning rate for each parameter. In addition to momentum accumulation, the algorithm considered the average non-centered variance of the gradients, which reduced the number of iterations required to converge the algorithm while improving the neural network’s prediction accuracy. The Adam algorithm is distinguished by using exponential moving averages for gradients and their squares, controlled by the parameters β₁ (for the gradient) and β₂ (for the gradient square). Selecting these averages determines the speed and accuracy of algorithm convergence. To solve the problem empirically, the values β₁ = 0.99 and β₂ = 0.999 were selected at the initial learning rate η = 10⁻³.

Continuous differentiability is a general requirement for an activation function, which comes from the requirements of algorithms based on calculation of the loss function gradient. The selection of the activation function type is typically influenced by the nature of the problem to be solved, the input and output data types, and the desired outcome of the neural network model. A linear activation function, for example, involves solving a linearly separable problem that does not require the hidden layer of the neural network. Hidden layers in a neural network indicate that a nonlinear separable problem is being solved. Nonlinear activation functions, such as sigmoidal functions, ReLU (Rectified Linear Unit) and its numerous varieties (Leaky ReLU, PReLU, RReLU), exponential functions, and radial basis functions, are required for such problems. Because of its advantages, such as monotonicity, boundedness, and the presence of a section close to linear, the sigmoidal activation function is convenient to use for the problem formulated here:

$$f\left(x,\alpha \right)=\frac{1}{1+e^{-\alpha x}}, \forall \alpha >0,$$

(2)

where the parameter α = 0.5 corresponds to a large length of the nearly linear function section. In Equation (2), the parameter α is responsible for the slope of the linear section near the ordinate 0. At α = 0, the sigmoid function turns into a line y = 0.5; for α = + ∞, it turns into a threshold function. The choice of the parameter α = 0.5, or another value close to 0, leads to an increase in the length of the “linear” section of the function, as a result of which the sigmoid function behaves like a linear function (in the working section), while maintaining its nonlinearity.

In turn, the mapping from the associative layer space to the response layer space is linear, meaning that the value of the response layer neuron is the weighted sum of the values of the associative layer neurons.

The Euclidean norm between the target vector and the neural network’s response vector is the way to tell if the network is getting closer to the target value:

$$d\left(p,q\right)=\sqrt{{\displaystyle \sum _{i=0}^{K-1}{\left(p_i-q_i\right)}^2}} ,$$

(3)

where p is the target vector, q is the response vector of the neural network, and K is the dimension of the target vector p.

4. Training

By using a mathematical model that generates real-time input data at each training phase, the accumulation of a collection of training and test data in computer memory can be avoided. Furthermore, this method eliminates the need to label training and test datasets. Parameters from the ranges listed in

Table 2. FBG reflection spectrum modeling parameters.

Parameters	Values	Normalized Values
FBG length (L)	0.5 ÷ 3.5 mm	-
Refractive index (n)	1.4586604	-
Induced refractive index (Δn)	0.5∙10−4 ÷ 1.5∙10−4	-
Controlled wavelength interval (Δ)	1548.5 ÷ 1551.5 nm	0 ÷ 1
Central wavelength (λB)	1549.5 ÷ 1550.5 nm	0.333 ÷ 0.667
Discretization interval (Δλ)	0.16667 nm	1/18
Full width at half height	0.229 ÷ 1.458 nm	0.076 ÷ 0.486
Maximum amplitude	0.61 × 216 a.u.	1
Amplitude variation range	3.92∙10−4 × 216 ÷ 0.15 × 216 a.u.	6.42∙10−4 ÷ 0.26
Noise	0 ÷ 2 × 10−3 × 216 a.u.	0 ÷ 0.002
Pedestal	0 ÷ 2 × 10−1 × 216 a.u.	0 ÷ 0.2

were used to create a model of the input data.

The algorithm that underpins the neural network training process includes the series of steps below:

• Modeling of the FBG reflection spectrum at a wavelength arbitrarily selected from the range [λ_B − 0.5 nm; λ_B + 0.5 nm], where λ_B is the central value of the wavelength within the controlled wavelength interval Δ. The resulting spectrum {R_i} is discretized at Δλ intervals. The amplitude values {R_i} and the central wavelength λ_B are normalized by (1).
• Setting the initial values of synaptic weights and shifts:

  $$w_{k,j}^n=\frac{1}{2}-\mathrm{rnd}\left(1\right), b_k^n=\frac{1}{2}-\mathrm{rnd}\left(1\right) .$$

  (4)
• Direct signal propagation through a neural network, consisting of these steps:
  • Calculating the weighted sum of the sensory layer neurons’ values;
  • Calculating the associative layer values (using a nonlinear activation function);
  • Calculating the weighted sum of the associative layer’s neurons’ values;
  • Calculating the value of the neurons in the response layer (using a linear activation function) for each neuron in the associative and reacting layers.
  • Direct propagation yielding a normalized FBG central wavelength value representing neural network responses.
• Reverse propagation of the error signal, consisting of these steps:
  • Calculating the error of the FBG central wavelength determination (the norm between the target value vector and the neural network response vector) according to the selected metric;
  • Calculating the error for each neural network parameter in sequence using the Adam optimization algorithm;
  • Correcting synaptic weights and model shifts.
• Verifying that the training algorithm criterion meets the requirement:

  $$\left|\frac{1}{I}{\displaystyle \sum _{i=K-I}^{K-1}\mathrm{d}\left({\lambda }_B^i,{\overline{\lambda }}^i\right)}-\frac{1}{I}{\displaystyle \sum _{j=K-2I}^{K-I-1}\mathrm{d}\left({\lambda }_B^j,{\overline{\lambda }}^j\right)}\right|<{\epsilon }_1,$$

  (5)

  where ${\lambda }_{\mathrm{B}}^{i(j)}$ and ${\overline{\lambda }}^{i(j)}$ are the target and calculated values of the FBG central wavelength at the i(j)th iteration, I is the number of iterations for which averaging occurs, K is the number of training iterations, and ε₁ is an arbitrary predetermined small value.

The neural network training that had occurred before the criterion met the requirement (5) amounted to a little more than 10⁶ training algorithm iteration cycles. Figure 3 shows the dependence of the absolute error in determining the FBG central wavelength found according to norm (3) on the number of training process cycles. The dependence in Figure 3 is plotted on a semi-logarithmic scale for better visualization. The error value begins to slow after 10⁶ training cycles and stabilizes at around 0.525 pm.

Neural network training generates a 10 × 19 matrix of weighting coefficients for the associative layer w_A, a 1 × 10 matrix for the response layer w_R, a 10 × 1 matrix for neuron shift vectors of associative b_A layers, and a 1 × 1 matrix for neuron shift vectors of response b_R layers.

The matrices of the weighting coefficients w_A, w_R, together with the shift vectors b_A and b_R, make it possible to determine the FBG reflection spectrum central wavelength according to the decisive rule, using the values of the 19 × 1 FBG reflection spectrum R as input data,

$$\overline{\lambda }=w_R\times f\left(w_A\times R+b_A,\alpha \right)+b_R,$$

(6)

and ensure that the FBG central wavelength value is restored from the spectral data (Figure 4) obtained under low-resolution conditions.

5. Verification

To verify the neural network model, we used an algorithm consisting of a series of steps similar to the training algorithm, except for the error back propagation part of the algorithm:

1.

Modeling of the FBG reflection spectrum at a wavelength arbitrarily selected from the range [λ_B − 0.5 nm; λ_B + 0.5 nm], where λ_B is the central value of the wavelength within the controlled wavelength interval Δ. The resulting spectrum {R_i} is discretized at Δλ intervals. The amplitude values {R_i} and the central wavelength λ_B are normalized using (1).

2.

Calculation of the neural network response $\overline{\lambda }$ using (6).

3.

Calculation of the error in FBG central wavelength determination using (3).

Figure 5 shows the neural network response’s dependence on the values of the FBG central wavelength λ for 10⁴ iterations of the verification algorithm. The red line represents the dependence corresponding to the neural network model’s response; the blue line (along the additional axis) represents the magnitude of the error in the central wavelength determination $\left(\lambda -\overline{\lambda }\right)$.

When the sensor system was in real-time operation, the profile of the FBG reflection spectrum serving as a sensor sensing element hardly changed. Consequently, it was intriguing to investigate the magnitude of the absolute error in determining the central wavelength that resulted from a plane-parallel shift of the same spectrum across the entire controlled wavelength section. Two approaches were of interest here. The first involved determining the FBG reflection spectrum central wavelength in the wavelength range with an interval Δ and fixed left λ_L and right λ_R boundaries. The second approach involved determining the FBG reflection spectrum central wavelength in the wavelength range with an interval Δ but with variable left λ_L and right λ_R boundaries. Here the boundary was determined according to a rule that ensured that the wavelength λ_Max corresponding to the maximum value in the discretely obtained spectrum λ_Max = max(R) was always located in the center of the wavelength range [λ_L, λ_R]. Figure 6 and Figure 7 show the absolute errors in determining wavelengths using the first and second range control approaches, respectively.

Comparing the errors in determining the central wavelength with fixed (Figure 6) and variable (Figure 7) boundaries revealed that the errors in the latter case were smaller and had an almost periodic dependence on the location of the FBG reflection spectrum maximum. It is also worth noting that the quasi-periodic dependence of errors in Figure 7 was caused by the errors belonging to the interval [1549.917, 1550.083] nm in Figure 6 (green area).

The distribution density of errors in determining the FBG central wavelength for each approach (fixed and variable boundaries) is shown in Figure 8 and Figure 9, respectively. In addition, the value of the FBG central wavelength error corresponding to the central wavelength shift for λ_B is plotted on the abscissa axis (on the additional ordinate axis).

The distribution of errors for both wavelength range boundary selection approaches was close to the normal distribution. For the fixed-boundary approach, the mathematical expectation was μ = 0.084 pm, and the standard deviation was σ = 0.627 pm, with most of the error values (lower and upper quartiles) belonging to the interval [−0.28, 0.46] pm. For variable boundaries, the mathematical expectation was μ = 0.045 pm, and the standard deviation was σ = 0.528 pm, with most error values belonging to the interval [−0.24, 0.39] pm. The statistical characteristics of the error distribution in calculating the central wavelength are shown in

Table 3. Statistical characteristics of the error distribution.

Characteristic	Value (Fixed Boundaries)	Value (Variable Boundaries)
Mathematical expectation	0.084 pm	0.041 pm
Average absolute error	0.478 pm	0.411 pm
Median	0.097 pm	0.066 pm
Standard deviation	0.627 pm	0.533 pm

.

6. Conclusions

The results demonstrated that a neural network based on a three-layer Rosenblatt’s perceptron could be successfully applied in fiber optic sensor problems, specifically when it was necessary to determine the central wavelength shift of a fiber Bragg grating used as a sensing element in a discrete sensor. The proposed method of employing a neural network had the advantages of requiring no complex calculations in the measurement mode and being implementable on simple microcontrollers. Furthermore, significant computational complexity was only incurred once during neural network training. During training, the proposed method transformed the spectrum values into dimensionless variables by shifting the coordinate axis. Once the calculations were completed, the dimensionless variables were converted back into dimensional values. This enabled a trained neural network to determine the FBG central wavelength across a broad range of wavelengths and for varying-quality FBGs. The suggested algorithm increased the resolution of wavelength shift determination by almost three orders of magnitude compared with the initial resolution of the spectrum analyzer, while maintaining the mathematical framework’s apparent simplicity. This ensured that a spectrum analyzer with a 0.167 nm resolution could determine the shift of the fiber Bragg grating spectrum with an error of no more than 0.5 pm, which further determined the resolution when measuring, for instance, a temperature of 0.05 °C. The advantages of the method included its independence from broadband light source power fluctuations and the ability to determine the central wavelengths of the FBGs with different quality factors and in different spectral ranges using a neural network trained only once.

The spectra used for training suggest apodized FBGs whose side lobes are suppressed, which corresponds to physically formed FBGs according to Lloyd’s recording scheme. Point-by-point writing [45,46] implies the possibility of recording non-apodized FBGs and FBGs of an arbitrary structure. The use of a neural network to determine the central wavelengths of structured FBGs or FBGs with phase or other inhomogeneities requires additional research. At the same time, the results obtained in this study make it possible to determine with certainty the center of wavelengths of non-overlapping apodized FBGs.

The main limitation of the proposed method was the impossibility of the central wavelength determination of the FBG with FWHM of less than 200 pm (~1.5 larger than the spectrum resolution). This limitation was due to the fact that when FWHM was less than the specified value, the FBG reflection spectrum obtained by the I-MON-512-based interrogator had one to two value points, which did not allow the determination of the value of the central wavelength with a given (required) accuracy. At the same time, there was a limitation in the impossibility of determining the central wavelength of the FBG with FWHM greater than the specified interval for finding the FBG (Δ = 3 nm). The third limitation was the impossibility of determining the central wavelengths of two or more FBGs in the wavelength section under consideration (Δ = 3 nm).

It must be noted that the second constraint is not a fundamental limitation of the proposed method, and in order to mitigate it, it is enough to retrain the neural network for a wider interval. The first limitation, however, can be removed only by increasing the physical resolution of the spectrum analyzer. Further research will be dedicated to the development of approaches for the central wavelength determination of two or more FBGs with overlapping spectra.