Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations

Abstract

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields.

1. Introduction

In recent years, fractional operators have emerged as a powerful mathematical tool for modeling complex systems [1,2,3,4], offering superior capabilities over integer-order calculus in capturing memory effects, hereditary properties, and non-local dynamics with greater accuracy. By incorporating non-local memory effects and capturing anomalous dynamics [5,6], fractional-order models provide a more accurate and flexible framework for describing phenomena such as viscoelasticity, diffusion in heterogeneous media, and biological processes with long-range dependencies. Their ability to characterize multi-scale behaviors and fit real-world data more precisely makes them invaluable in physics, engineering, and biology. Nevertheless, solving fractional partial differential equations (fPDEs) presents significantly greater complexity. The development of effective numerical methods to approximate fPDEs has been the goal of some researchers. The traditional solving methods include finite-difference method [7,8], finite element method [9,10], spectral method [11,12] and virtual-element method [13,14], etc. However, these algorithms require discretization, which incurs significant time costs due to the vast amount of data processing and introduces approximate errors.

The rapid development of artificial intelligence has driven widespread applications of deep learning in scientific and technological domains [15,16,17,18]. This trend is primarily propelled by the strong function approximation capability [19,20,21] of neural networks (NNs), which has established them as powerful computational tools for solving differential and integral equations [22,23]. Particularly in computational mathematics, physics-informed neural networks (PINNs) have emerged as a significant methodology for solving partial differential equations (PDEs), attracting considerable scholarly attention [24,25,26,27]. PINNs are machine learning models that combine deep learning with physical knowledge. By serving as a surrogate model, NNs establish a mapping between spatio-temporal points and the corresponding PDE solutions. The residual information of PDEs is embedded into the loss function of NNs, thereby optimizing the training process through minimization with respect to the parameters of NNs, ultimately leading to the derivation of an optimal model. Due to the inapplicability of the standard chain rule in fractional calculus, automatic differentiation cannot be directly used with fractional operators. To solve this problem, fractional physics-informed neural networks (fPINNs) [28,29,30] discretize the fractional operators numerically. By employing automatic differentiation for integer-order operators and numerical approximation for fractional operators, the residual information from fPDEs can be integrated into the NN’s loss function during training. However, fPINNs require numerical discretization to solve fPDEs, which inherently introduces approximation errors. The fPINNs are data-driven models [31,32,33] that require a large number of training points for training and have high time costs. During network training, the NNs often become stuck in the local optimal value and cannot find the global optimal value. Consequently, enhancing NN training efficiency and developing advanced optimization algorithms remain active research priorities.

Nevertheless, existing approaches universally employ NNs to compute numerical approximations for diverse classes of fPDEs. While numerical solutions of fPDEs have broad applicability in practical applications, analytical solutions still possess irreplaceable value in pursuing accuracy and theoretical depth. Generally, it becomes difficult to find exact analytical solutions of fPDEs. In prior work [34,35,36], we developed an NN framework that derives exact PDE solutions through equation-specific trial functions constructed via NN architectures. Currently, NNs have not yet been employed to obtain exact analytical solutions of fPDEs.

Motivated by these advances, a novel artificial neural networks-based analytical solver is newly introduced herein, addressing fPDEs through an unprecedented computational paradigm. This novel approach integrates NNs with a symbolic computation strategy, enabling the rapid acquisition of precise analytical solutions to fPDEs. The proposed method solely employs the feedforward computation of NNs to obtain exact solutions for fPDEs without involving any training mechanisms, thereby significantly improving computational efficiency and accuracy. Moreover, this analytical method exhibits high customizability and flexibility, allowing the construction of diverse trial functions by adjusting the parameter configuration of NNs, making it widely applicable to various types of fPDEs. The main focus of our contributions lies in the following aspects:

• We introduce an innovative framework for deriving exact analytical solutions to fPDEs, ensuring mathematically rigorous results free from computational errors.
• Potential analytical solutions of fPDEs are constructed via NN architectures. Transformed inputs undergo feedforward propagation to yield network outputs, which subsequently serve as trial functions in the fPDE solutions framework.
• Our approach simplifies the fPDE into computationally feasible algebraic systems through trial function application. The synaptic weights and biases of NNs are then resolved through undetermined coefficient optimization.
• Exact analytical solutions for fPDEs are obtained in a data-independent manner through this computational framework. NNs are employed to impose structural constraints on trial function formulation, enhancing mathematical tractability.

The paper adopts the following structure: Theoretical principles of the proposed solver are formalized within the Section 2. In Section 3, the accuracy and feasibility of the artificial neural networks-based analytical solver for fPDEs are verified through fractional wave equation [37], fractional telegraph equation [38], fractional Sharma–Tasso–Olever equation [39], and fractional biological population model [40]. In Section 4, the proposed method is discussed. Finally, the conclusions of this paper are given in Section 5.

2. Methodology

In this section, we introduce the idea of an artificial neural network-based analytical solver for finding solutions of fPDEs. Analytical solutions of fPDEs are obtained by embedding NNs as explicit functional representations. The novelty of the technique stems from its ability to impose explicit mathematical constraints on the solution space representation. Consider the following general fractional partial differential equation

$$P(u,D_t^{\alpha }u,D_x^{\beta }u,D_t^{2\alpha }u,D_x^{2\beta }u,\dots )=0,$$

(1)

where the function $u(x,t)$ satisfies the governing equation, the fractional orders $0<\alpha \le 1$ and $0<\beta \le 1$, $D_t^{\alpha }u$ and $D_x^{\beta }u$ are the conformable fractional derivatives of $u(x,t)$ with respect to t and x, respectively.

We employ a precisely formulated explicit model during the forward propagation phase of the NNs to compute the solution $u(x,t)$ for Equation (1). The neuron model is composed of input, output, and computational functions. The input can be analogized as dendrites of neurons, while the output can be analogized as axons of neurons, and the computing can be analogized as the nucleus. Figure 1 shows a neuron model with three inputs, one output, and two computational functions. The arrow lines in Figure 1 are called connections, and each connection has a weight on it. In this networks, the weighted sum of the three inputs $c_1$, $c_2$, and $c_3$ is first computed as $sum=w_1{c}_1+w_2{c}_2+w_3{c}_3$. Subsequently, they go through a nonlinear transformation to obtain

$$z=f\left(sum\right)=f(w_1{c}_1+w_2{c}_2+w_3{c}_3+b),$$

(2)

here, f denotes the activation function, enhancing the neuron’s ability to model nonlinear relationships, while b represents the bias term. Consequently, the solution to the equation can be explicitly represented by a single neuron’s output. Given the high expressive power of neural networks in approximating functions, solutions to fPDEs can be computed using NNs with limited neurons or hidden layers. In this work, we employ an NN-based model to approximate the solution $u(x,t)$ of the fPDEs, as illustrated in Figure 2. Where X and T are inputs of the NNs, and

$$X={\displaystyle \frac{x^{\beta }}{\beta }},T={\displaystyle \frac{t^{\alpha }}{\alpha }},$$

(3)

After applying the aforementioned independent variable transformation, all fractional derivatives in Equation (1) are converted into integer-order derivatives

$$P(u,u_T,u_X,u_{TT},u_{XX},\dots )=0,$$

(4)

thus simplifying the subsequent calculation. Consider an NN architecture comprising n hidden layers, with each layer containing m neurons. For the first hidden layer ($L_1$), the output of its n-th neuron ($z_{1n}$) can be expressed as

$$z_{1n}=f\left(sum\right)=f(w_{xn}X+w_{tn}T),$$

(5)

where f is the activation function, while $w_{xn}$ and $w_{tn}$ denote the weights linking the n-th neuron to inputs X and T, respectively. The output $z_{nn}$ of the n-th neuron in layer ($L_n$) can be expressed as

$$z_{nn}=f\left(sum\right)=f(w_{1n}^{n-1}{z}_{(n-1)1}+\dots +w_{mn}^{n-1}{z}_{(n-1)m}),$$

(6)

let $z_{(n-1)1}$ denote the activation value of the first neuron in the $(n-1)$-th hidden layer, and $z_{(n-1)m}$ represent the output of the m-th neuron in the same layer. The connection weight from neuron $z_{(n-1)1}$ to $z_{nn}$ is denoted as $w_{1n}^{n-1}$, whereas $w_{mn}^{n-1}$ represents the weight between $z_{(n-1)m}$ and $z_{nn}$. The system’s final output is computed as the weighted linear combination of all activation outputs from the final layer’s neurons. Consequently, the potential closed-form solution $u(x,t)$ of the fPDEs, derived via the NNs framework, takes the following form:

$$u(x,t,w)=w_{1u}{z}_{n1}+w_{2u}{z}_{n2}+\dots +w_{mu}{z}_{nm}.$$

(7)

The trial function takes the expanded representation $u(x,t,w,b)$ when neuronal outputs contain bias components.

In the realm of mathematical modeling for physical phenomena, utilizing the trial function method to derive analytical solutions for field partial differential equations related to these domains is a prevalent technique. Nevertheless, the cornerstone of this methodology lies in the artful construction of the trial function itself, which necessitates a meticulous and strategic construction process. This paper presents an analytical solver based on an NN architecture, employing its explicit mathematical formulation as the trial solution for fPDEs. The architecture of the NNs is designed to be regular, and the overall flow of the proposed method is shown in Figure 3.

The method uses the trial function method to construct potential analytical solutions of fPDEs. The potential analytical solutions to fPDEs may incorporate various elementary functions, including rational term ($1/(·)$), trigonometric components ($sin(·)$, $cos(·)$), and exponential term (${\mathrm{e}}^{(·)}$), etc. This is directly related to the activation function of the selected NNs. Fortunately, the selection of the activation function can be facilitated by leveraging the initial or boundary conditions as a guiding principle, significantly expediting the process of designing the architecture of NNs. The NNs have a structured architecture with a standardized pattern, mainly requiring specification of hidden layer neuron quantities and individual neuron activation functions, as detailed in

Table 1. Architectural configuration of NNs for analytical solution.

Architectural Component	Configuration Details
Network Depth	Number of hidden layers
	Neuronal distribution across layers
Node Characteristics	Nonlinear activation selection
	Bias term implementation
Output Processing	Linear combination of weighted inputs

. Consequently, the proposed methodology significantly reduces the difficulties involved in formulating trial functions.

Suppose there is an architecture of NNs with one hidden layer and two neurons, and $sin(·)$ is chosen as the activation function for each neuron. If $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$ are taken as inputs to the NNs, the explicit model $u(X,T,w,b)$ of the NNs can be expressed as

$$u(X,T,w,b)=w_{1u}sin(w_{x1}X+w_{t1}T+b_1)+w_{2u}sin(w_{x2}X+w_{t2}T+b_2)+b_3,$$

(8)

the connection weights between neurons are denoted by w, while b corresponds to the bias term in each neuron’s output. By substituting the Equation (8) into the Equation (4), a nonlinear equation about weights and biases is obtained:

$$\mathrm{Eqs}(X,T,w,b)=0,$$

(9)

by combining the similar terms of Equation (9), extracting the coefficients, and setting all coefficients to zero, we derive a system of nonlinear algebraic equations. Finally, the weight coefficients, biases, and Equation (3) obtained by the solution are substituted into the NN model Equation (8), and the analytical solution $u(x,t)$ of the Equation (1) can be obtained.

To facilitate the practical application of our analytical methodology, we have implemented a dedicated Maple-based computational tool that automates all solution procedures.

3. Applications

In this section, we demonstrate the accuracy and feasibility of an artificial neural network-based analytical solver to solve fPDEs by fractional wave equation [37], fractional telegraph equation [38], fractional Sharma–Tasso–Olever equation [39], and fractional biological population model [40].

3.1. Fractional Wave Equation

The fractional wave equation under consideration takes the form

$$D_t^{2\alpha }u(x,t)=k^2{D}_x^{2\beta }u(x,t),$$

(10)

with the following initial condition

$$u(x,0)=2x^2+x,$$

(11)

where k is a positive coefficient, $x\in R$, $t\in R^{+}$, $1<2\alpha \le 2$, $1<2\beta \le 2$, the function $u(x,t)$ satisfies the governing equation.

Via the variable transformation Equation (3), Equation (10) is rewritten as its integer-order counterpart:

$$u_{TT}=k^2{u}_{XX},$$

(12)

Case I: To obtain analytical solutions for the fractional wave equation, the NNs architecture is utilized. Design considerations incorporate the initial condition of the equation. We implement a network with two hidden layers, each consisting of two neurons. The corresponding architectural schematic appears in Figure 4a. Using neural networks, the trial function is parameterized by weights and biases. Therefore, the mathematical representation of the explicit model can be expressed as follows:

$$\left\{\begin{array}{c}{N}_1=T{w}_{t1}+X{w}_{x1}+b_1,\hfill \\ N_2=T{w}_{t2}+X{w}_{x2}+b_2,\hfill \\ N_3=w_{23}{f}_2\left(N_2\right)+w_{13}{f}_1\left(N_1\right)+b_3,\hfill \\ N_4=w_{24}{f}_2\left(N_2\right)+w_{14}{f}_1\left(N_1\right)+b_4,\hfill \\ u(X,T)=w_{3u}{f}_3\left(N_3\right)+w_{4u}{f}_4\left(N_4\right)+b_5,\hfill \end{array}\right.$$

(13)

where $X=x^{\beta }/\beta$, $T=t^{\alpha }/\alpha$, $u(X,T)$ denote the exact solution of the Equation (10). In the networks architecture, the initial hidden layer generates outputs $N_1$ and $N_2$ from its first and second neurons, respectively, followed by outputs $N_3$ and $N_4$ from the subsequent layer’s corresponding neurons, $w_{t1}$, $w_{t2}$, $w_{x1}$, $w_{x2}$, $w_{13}$, $w_{23}$, $w_{14}$, $w_{24}$, $w_{3u}$, $w_{4u}$ are the weights between the each neurons, $f_1$, $f_2$, $f_3$, $f_4$ are the activation functions. The bias terms are denoted as $b_1$, $b_2$, $b_3$, $b_4$, and $b_5$, corresponding to each neuron’s offset parameter. Within the NNs framework, the trial solution $u(x,t)$ for Equation (13) is parameterized by both connection weights and these bias components.

The first hidden layer employs exponential functions ${\mathrm{e}}^{(·)}$ for all neuronal activations, while in the second hidden layer, the third and fourth neurons utilize $(·)$ and ${(·)}^2$ activation mappings, respectively. Consequently, the explicit model $u(X,T)$ admits the following mathematical representation:

$$\begin{array}{cc}\hfill u(X,T)=& {\left(w_{14}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}+b_4\right)}^2{w}_{4u}\hfill \\ \hfill & +\left(w_{13}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{23}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}+b_3\right)w_{3u}+b_5.\hfill \end{array}$$

(14)

By substituting the NNs-based Equation (14) into the Equation (12), we obtain

$$\begin{array}{cc}\hfill & 2w_{4u}{\left(w_{14}{w}_{t1}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{w}_{t2}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\right)}^2\hfill \\ \hfill & +2w_{4u}\left(w_{14}{\mathrm{e}}^{T{w}_{t,1}+X{w}_{x1}+b_1}+w_{24}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}+b_4\right)\hfill \\ \hfill & \times \left(w_{14}{w}_{t1}^2{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{w}_{t2}^2{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\right)\hfill \\ \hfill & +w_{3,u}\left(w_{13}{w}_{t1}^2{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{23}{w}_{t2}^2{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\right)\hfill \\ \hfill & -k^2\left[2w_{4u}{\left(w_{14}{w}_{x1}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{w}_{x2}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\right)}^2\right.\hfill \\ \hfill & +2w_{4u}\left(w_{14}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}+b_4\right)\hfill \\ \hfill & \times \left(w_{14}{w}_{x1}^2{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{w}_{x2}^2{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\right)\hfill \\ \hfill & \left.+w_{3u}\left(w_{13}{w}_{x1}^2{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{23}{w}_{x2}^2{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\right)\right]\hfill \\ \hfill & =0.\hfill \end{array}$$

(15)

By collecting terms involving the basis set $\{X,T,{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1},{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\}$ and enforcing zero coefficients for each independent component, we derive the following equation system.

$$\left\{\begin{array}{l}-2k^2{b}_4{w}_{14}{w}_{4u}{w}_{x1}^2-k^2{w}_{13}{w}_{3u}{w}_{x1}^2+2b_4{w}_{14}{w}_{4u}{w}_{t1}^2+w_{13}{w}_{3u}{w}_{t1}^2=0,\\ -2k^2{b}_4{w}_{24}{w}_{4u}{w}_{x2}^2-k^2{w}_{23}{w}_{3u}{w}_{x2}^2+2b_4{w}_{24}{w}_{4u}{w}_{t2}^2+w_{23}{w}_{3u}{w}_{t2}^2=0,\\ -2k^2{w}_{14}{w}_{24}{w}_{4u}{w}_{x1}^2-4k^2{w}_{14}{w}_{24}{w}_{4u}{w}_{x1}{w}_{x2}-2k^2{w}_{14}{w}_{24}{w}_{4u}{w}_{x2}^2\\ +2w_{14}{w}_{24}{w}_{4u}{w}_{t1}^2+4w_{14}{w}_{24}{w}_{4u}{w}_{t1}{w}_{t2}+2w_{14}{w}_{24}{w}_{4u}{w}_{t2}^2=0,\\ -4k^2{w}_{14}^2{w}_{4u}{w}_{x1}^2+4w_{14}^2{w}_{4u}{w}_{t1}^2=0,\\ -4k^2{w}_{24}^2{w}_{4u}{w}_{x2}^2+4w_{24}^2{w}_{4u}{w}_{t2}^2=0.\end{array}\right.$$

(16)

Through the systematic solving of the equation system, we derive 20 constraint conditions, where each condition corresponds to a general solution in the solution space. Due to limited space, a few representative examples are cited below:

Solution 1:

$$\begin{array}{c}\left\{k=k,b_4=b_4,w_{13}=w_{13},w_{14}=0,w_{23}=w_{23},w_{24}=w_{24},w_{3u}=0,\right.\hfill \\ \left.w_{4u}=w_{4u},w_{t1}=w_{t1},w_{t2}=k{w}_{x2},w_{x1}=w_{x1},w_{x2}=w_{x2}\right\}.\hfill \end{array}$$

(17)

The general solution for this system of constraints can be expressed as follows:

$$u(X,T)={\left(w_{24}{\mathrm{e}}^{Tk{w}_{x2}+X{w}_{x2}+b_2}+b_4\right)}^2{w}_{4u}+b_5.$$

(18)

Substituting $X=x^{\beta }/\beta$, $T=t^{\alpha }/\alpha$ into the Equation (18) yields the exact solution

$$u_1(x,t)={\left(w_{24}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x2}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x2}}{\beta }}+b_2}+b_4\right)}^2{w}_{4u}+b_5.$$

(19)

Solution 2:

$$\begin{array}{c}\left\{k=k,b_4=b_4,w_{13}=0,w_{14}=0,w_{23}=w_{23},w_{24}=w_{24},w_{3u}=w_{3u},w_{4u}=w_{4u},\right.\hfill \\ \left.w_{t1}=w_{t1},w_{t2}=k{w}_{x2},w_{x1}=w_{x1},w_{x2}=w_{x2}\right\},\hfill \end{array}$$

(20)

and for the given constraints, the general solution $u(x,t)$ takes the following form:

$$\begin{array}{cc}\hfill u_2(x,t)=& {\left(w_{24}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x2}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x2}}{\beta }}+b_2}+b_4\right)}^2{w}_{4u}+\left(w_{23}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x2}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x2}}{\beta }}+b_2}+b_3\right)w_{3u}+b_5.\hfill \end{array}$$

(21)

Solution 3:

A fully connected NN structure can better approximate the analytical solution of an equation. However, the weight coefficients in the constraint conditions corresponding to the three examples introduced above are not all non-zero, which means the corresponding NNs are not fully connected network structures. For example, the network structure corresponding to the set of constraints Equation (17) of the equation only utilizes the second and fourth neurons. Due to limited space, we provide the constraint conditions corresponding to a fully connected NN structure as follows

$$\begin{array}{c}\left\{k=k,b_4=b_4,w_{13}=w_{13},w_{14}=w_{14},w_{23}=w_{23},w_{24}=w_{24},w_{3u}=w_{3u},\right.\hfill \\ \left.w_{4u}=w_{4u},w_{t1}=k{w}_{x1},w_{t2}=k{w}_{x2},w_{x2}=w_{x2}\right\},\hfill \end{array}$$

(22)

and the general solution for this system of constraints can be expressed as follows:

$$\begin{array}{cc}\hfill u_3(x,t)& ={\left(w_{14}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x1}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x1}}{\beta }}+b_1}+w_{24}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x2}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x2}}{\beta }}+b_2}+b_4\right)}^2{w}_{4u}\hfill \\ \hfill & +\left(w_{13}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x1}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x1}}{\beta }}+b_1}+w_{23}{\mathrm{e}}^{{\displaystyle \frac{t^{\alpha }k{w}_{x2}}{\alpha }}+{\displaystyle \frac{x^{\beta }{w}_{x2}}{\beta }}+b_2}+b_3\right)w_{3u}\hfill \\ \hfill & +b_5.\hfill \end{array}$$

(23)

In order to further simplify the general solution of the Equation (23), we substitute the initial condition into Equation (23) to obtain

$$\begin{array}{cc}\hfill u(x,t)=& {\left(w_{14}{\mathrm{e}}^{Tk{w}_{x1}+X{w}_{x1}+b_1}+w_{24}{\mathrm{e}}^{Tk{w}_{x2}+X{w}_{x2}+b_2}+b_4\right)}^2{w}_{4u}\hfill \\ \hfill & +\left(w_{13}{\mathrm{e}}^{Tk{w}_{x1}+X{w}_{x1}+b_1}+w_{23}{\mathrm{e}}^{Tk{w}_{x2}+X{w}_{x2}+b_2}+b_3\right)w_{3u}\hfill \\ \hfill & -\left({\mathrm{e}}^{2(X{w}_{x1}+b_1)}{w}_{14}^2+2{\mathrm{e}}^{X{w}_{x1}+b_1}{\mathrm{e}}^{X{w}_{x2}+b_2}{w}_{14}{w}_{24}+{\mathrm{e}}^{2(X{w}_{x2}+b_2)}{w}_{24}^2\right)w_{4u}\hfill \\ \hfill & -2\left({\mathrm{e}}^{X{w}_{x1}+b_1}{b}_4{w}_{14}+{\mathrm{e}}^{X{w}_{x2}+b_2}{b}_4{w}_{24}\right)w_{4u}-b_4^2{w}_{4u}+2X^2-b_3{w}_{3u}+X\hfill \\ \hfill & -\left({\mathrm{e}}^{X{w}_{x1}+b_1}{w}_{13}+{\mathrm{e}}^{X{w}_{x2}+b_2}{w}_{23}\right)w_{3u},\hfill \end{array}$$

(24)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

If one chooses coefficients $\{k=1,b_1=0.1,b_2=0.1,b_3=0.1,b_4=0.1,w_{13}=0.5,w_{14}=1,w_{23}=1,w_{24}=1,w_{3u}=1,w_{4u}=0.1,w_{x1}=0.1,w_{x2}=0.5,\alpha =0.6,\beta =0.6\}$ in the Equation (24), the results are shown in Figure 5. In this way, the solution’s performance becomes more apparent. Figure 5a presents a 3D plot illustrating the solution over the spatio-temporal domain $[0,5]\times [0,5]$. To observe the local behavior of the solution, we plot the x-curves in Figure 5b at interval $t\in [0,5]$. In Figure 5c,d, we illustrate the contour and density representations of the fractional wave equation. As can be seen from the figure, the initial condition takes on a parabolic shape. Over time, the shape and size of the solution evolve, forming distinct peaks and troughs while exhibiting characteristics of propagation and diffusion. The variation in color intuitively reflects the dynamic changes in the solution, providing a crucial visualization tool for understanding the solutions to the fractional wave equation.

Case II: Therefore, the artificial neural networks-based analytical solver can effectively obtain the exact solutions of the Equation (10). Further demonstrating the adjustable nature of our approach, we employed a different NN parameter configuration for Equation (10), visible in Figure 4b. In this case, we take the activation functions of neurons in the second hidden layer as the $sin(·)$ and the $cos(·)$, respectively, while other settings in the NN structure remain unchanged. According to the analytical formulation Equation (13), the following trial function can be constructed:

$$\begin{array}{cc}\hfill u(X,T)=& {\left(w_{14}sin(T{w}_{t1}+X{w}_{x1}+b_1)+w_{24}cos(T{w}_{t2}+X{w}_{x2}+b_2)+b_4\right)}^2{w}_{4u}\hfill \\ \hfill & +\left(w_{13}sin(T{w}_{t1}+X{w}_{x1}+b_1)+w_{23}cos(T{w}_{t2}+X{w}_{x2}+b_2)+b_3\right)w_{3u}\hfill \\ \hfill & +b_5,\hfill \end{array}$$

(25)

Substituting Equation (25) into Equation (12) and combining similar terms involving $\{X,T,cos(T{w}_{t1}+X{w}_{x1}+b_1),cos(T{w}_{t2}+X{w}_{x2}+b_2),sin(T{w}_{t1}+X{w}_{x1}+b_1),sin(T{w}_{t2}+X{w}_{x2}+b_2)\}$. Setting every coefficient equal to zero generates 20 distinct constraint groups, each associated with a generalized solution of the equation that includes weights and biases. We give an example of the constraint conditions corresponding to a fully connected NN structure (where the weights are all non-zero), as shown below:

$$\begin{array}{c}\left\{k=k,b_4=b_4,w_{13}=w_{13},w_{14}=w_{14},w_{23}=w_{23},w_{24}=w_{24},w_{3u}=w_{3u},\right.\hfill \\ \left.w_{4u}=w_{4u},w_{t1}=k{w}_{x1},w_{t2}=k{w}_{x2},w_{x1}=w_{x1},w_{x2}=w_{x2}\right\},\hfill \end{array}$$

(26)

and the general solution $u(x,t)$ takes the form

$$\begin{array}{cc}\hfill u(x,t)=& {\left(w_{14}sin\left(\left(Tk+X\right)w_{x1}+b_1\right)+w_{24}cos\left(\left(Tk+X\right)w_{x2}+b_2\right)+b_4\right)}^2{w}_{4u}\hfill \\ \hfill & +\left(w_{13}sin\left(\left(Tk+X\right)w_{x1}+b_1\right)+w_{23}cos\left(\left(Tk+X\right)w_{x2}+b_2\right)+b_3\right)w_{3u}\hfill \\ \hfill & +b_5,\hfill \end{array}$$

(27)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

Substituting the Equation (11) into the Equation (27), we obtain

$$\begin{array}{cc}\hfill u(x,t)=& {\left(w_{14}sin\left(\left(Tk+X\right)w_{x1}+b_1\right)+w_{24}cos\left(\left(Tk+X\right)w_{x2}+b_2\right)+b_4\right)}^2{w}_{4u}\hfill \\ \hfill & +\left(w_{13}sin\left(\left(Tk+X\right)w_{x1}+b_1\right)+w_{23}cos\left(\left(Tk+X\right)w_{x2}+b_2\right)+b_3\right)w_{3u}\hfill \\ \hfill & -\left({sin}^2\left(X{w}_{x1}+b_1\right)\right)w_{14}^2{w}_{4u}-2sin\left(X{w}_{x1}+b_1\right)cos\left(X{w}_{x2}+b_2\right)w_{14}\hfill \\ \hfill & \times w_{24}{w}_{4u}-\left({cos}^2\left(X{w}_{x2}+b_2\right)\right)w_{24}^2{w}_{4u}-2sin\left(X{w}_{x1}+b_1\right)b_4{w}_{14}{w}_{4u}\hfill \\ \hfill & -2cos\left(X{w}_{x2}+b_2\right)b_4{w}_{24}{w}_{4u}-sin\left(X{w}_{x1}+b_1\right)w_{13}{w}_{3u}\hfill \\ \hfill & -cos\left(X{w}_{x2}+b_2\right)w_{23}{w}_{3u}-b_4^2{w}_{4u}+2X^2-b_3{w}_{3u}+X,\hfill \end{array}$$

(28)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

With the coefficients configuration $\{k=2,b_1=2,b_2=2,b_3=2,b_4=2,w_{13}=1,w_{14}=1,w_{23}=1,w_{24}=1,w_{3u}=1,w_{4u}=1,w_{x1}=2,w_{x2}=2,\alpha =0.6,\beta =0.6\}$ specified in Equation (28), the 3D plot, x-curves plot, contour lines, and density patterns of the Equation (10) are collectively illustrated in Figure 6. The Figure 6 exhibits a distinct wave-like pattern, indicating that the solution possesses wave characteristics in both space and time. The amplitudes of the wave peaks and troughs gradually change over time, revealing the dynamic evolution of the wave properties. In the spatial direction, the waveform displays a certain periodicity, suggesting that the solution exhibits periodic wave characteristics in space. In the Figure 6b, it is also evident that the phase of the curves shifts at different time points, indicating the presence of phase variation as the solution evolves dynamically over time.

3.2. Fractional Telegraph Equation

The following fractional telegraph equation is investigated:

$$D_t^{\alpha }u(x,t)+D_t^{2\alpha }u(x,t)+u(x,t)=D_x^{\beta }u(x,t),$$

(29)

with the following initial condition

$$u(x,0)=e^{-x},$$

(30)

where $0<\alpha \le 1$, $0<\beta \le 1$, $x\in R$, $t\in R^{+}$, the function $u(x,t)$ satisfies the equation.

Through the independent variable transformation Equation (3), Equation (29) is rewritten as

$$u_T+u_{TT}+u(x,t)=u_{XX},$$

(31)

Case I: Following the same approach, the NNs framework is employed to derive exact solutions for the fractional telegraph equation, where the initial conditions serve as the basis for designing the network structure. NN architecture 3, as shown in Figure 7a, is used to solve the equation. The network configuration consists of two hidden layers, both containing two neuronal units. Neurons in the initial hidden layer utilize exponential activation ${\mathrm{e}}^{(·)}$, whereas the subsequent layer adopts reciprocal activation $1/(·)$. According to the analytical formulation Equation (13), we can determine the trial function

$$\begin{array}{cc}\hfill u(X,T)=& {\displaystyle \frac{w_{3u}}{w_{13}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{23}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}+b_3}}\hfill \\ \hfill & +{\displaystyle \frac{w_{4u}}{w_{14}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1}+w_{24}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}+b_4}}+b_5,\hfill \end{array}$$

(32)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

Substituting Equation (32) into Equation (31) and grouping similar terms involving $\{X,T,{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+b_1},{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+b_2}\}$. Setting every coefficient equal to zero generates 18 distinct constraint groups, each associated with a generalized solution of the equation that includes weights and biases. We give an example of the constraint conditions corresponding to a fully connected NN structure, as shown below:

$$\begin{array}{c}\left\{b_3=0,b_4=0,b_5=0,w_{13}=w_{13},w_{14}=w_{14},w_{23}=w_{23},w_{24}=w_{24},w_{3u}=w_{3u},\right.\hfill \\ \left.w_{4u}=w_{4u},w_{t1}=w_{t2},w_{t2}=w_{t2},w_{x1}=-w_{t2}^2+w_{t2}-1,w_{x2}=-w_{t2}^2+w_{t2}-1\right\},\hfill \end{array}$$

(33)

and the analytical solution $u(x,t)$ takes the form

$$\begin{array}{cc}\hfill u(x,t)=& {\displaystyle \frac{w_{3u}}{w_{13}{\mathrm{e}}^{T{w}_{t2}+X\left(-w_{t2}^2+w_{t2}-1\right)+b_1}+w_{23}{\mathrm{e}}^{T{w}_{t2}+X\left(-w_{t2}^2+w_{t2}-1\right)+b_2}}}\hfill \\ \hfill & +{\displaystyle \frac{w_{4u}}{w_{14}{\mathrm{e}}^{T{w}_{t2}+X\left(-w_{t2}^2+w_{t2}-1\right)+b_1}+w_{24}{\mathrm{e}}^{T{w}_{t2}+X\left(-w_{t2}^2+w_{t2}-1\right)+b_2}}},\hfill \end{array}$$

(34)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$, and by substituting the Equation (30) into the above equation, we obtain the analytic solution in the following form

$$\begin{array}{cc}\hfill u(x,t)=& {\displaystyle \frac{\begin{array}{cc}\hfill & w_{14}^2{w}_{1,3}{\mathrm{e}}^{A+3b_1-b_2}+2w_{24}{w}_{13}{w}_{14}{\mathrm{e}}^{A+2b_1}+w_{23}{w}_{14}^2{\mathrm{e}}^{A+2b_1}\hfill \\ \hfill & +w_{24}^2{w}_{13}{\mathrm{e}}^{A+b_1+b_2}+2w_{24}{w}_{14}{w}_{23}{\mathrm{e}}^{A+b_1+b_2}+w_{24}^2{w}_{23}{\mathrm{e}}^{A+2b_2}\hfill \end{array}}{\begin{array}{cc}\hfill & \left({\mathrm{e}}^{b_1-b_2}{w}_{14}+w_{24}\right)\left(w_{13}{\mathrm{e}}^{B+b_1}+w_{23}{\mathrm{e}}^{B+b_2}\right)\left(w_{14}{\mathrm{e}}^{B+b_1}+w_{24}{\mathrm{e}}^{B+b_2}\right)\hfill \end{array}}},\hfill \end{array}$$

(35)

where $A=-2X{w}_{t2}^2+(T+2X)w_{t2}-3X$, $B=-X{w}_{t2}^2+(T+X)w_{t2}-X$, $X=x^{\beta }/\beta$, and $T=t^{\alpha }/\alpha$.

Case II: Furthermore, the NNs architecture 4 as shown in Figure 7b is used to solve the fractional telegraph equation. The network structure includes three hidden layers, and the activation functions of neurons in the first hidden layer, second hidden layer, and the third hidden layer are set to identity mapping $(·)$, ${\mathrm{e}}^{(·)}$, and reciprocal $1/(·)$, respectively. Following the architecture of the established NNs, we formulate the novel trial function as

$$\begin{array}{cc}\hfill u(X,T)=& {\displaystyle \frac{w_{5u}}{w_{13}{\mathrm{e}}^{A_1{w}_{13}+A_2{w}_{23}+b_3}+w_{23}{\mathrm{e}}^{A_1{w}_{14}+A_2{w}_{2,4}+b_4}+b_5}}\hfill \\ \hfill & +{\displaystyle \frac{w_{6u}}{w_{14}{\mathrm{e}}^{A_1{w}_{13}+A_2{w}_{23}+b_3}+w_{24}{\mathrm{e}}^{A_1{w}_{14}+A_2{w}_{24}+b_4}+b_6}}+b_7,\hfill \end{array}$$

(36)

where $A_1=T{w}_{t1}+X{w}_{x1}+b_1$, $A_2=T{w}_{t2}+X{w}_{x2}+b_2$, $X={\displaystyle \frac{x^{\beta }}{\Gamma (1+\beta )}}$ and $T={\displaystyle \frac{t^{\alpha }}{\Gamma (1+\alpha )}}$.

Compared with previously discussed trial functions, this formulation exhibits significantly greater complexity. Substituting Equation (36) into Equation (31) and grouping similar terms involving $\{X,T,{\mathrm{e}}^{A_1{w}_{13}+A_2{w}_{23}+b_3},{\mathrm{e}}^{A_1{w}_{14}+A_2{w}_{24}+b_4}\}$. When all coefficients are set to zero, the system yields 25 constraint groups, mapping to 18 weighted solutions of the equation with bias terms. We give an example of the constraint conditions corresponding to a fully connected NN structure, as shown below:

$$\begin{array}{c}\left\{b_5=0,b_6=0,b_7=0,w_{13}=w_{14},w_{14}=w_{14},w_{23}=w_{24},w_{24}=w_{24},w_{5u}=w_{5u},\right.\hfill \\ w_{x1}=-\left(w_{14}^2{w}_{t1}^2+2w_{14}{w}_{24}{w}_{t1}{w}_{t2}+w_{24}^2{w}_{t2}^2-w_{14}{w}_{t1}-w_{24}{w}_{t2}+w_{24}{w}_{x2}+1\right)\hfill \\ \left./w_{14},w_{6u}=w_{6u},w_{t1}=w_{t1},w_{t2}=w_{t2},w_{x2}=w_{x2}\right\}.\hfill \end{array}$$

(37)

Consequently, the exact solution satisfying these constraint conditions can be expressed as

$$\begin{array}{c}\hfill u(x,t)={\displaystyle \frac{\begin{array}{c}\hfill w_{5u}+w_{6u}\end{array}}{\begin{array}{c}\hfill w_{14}{\mathrm{e}}^{A_3+A_4+b_3}+w_{24}{\mathrm{e}}^{A_3+A_4+b_4}\end{array}}},\end{array}$$

(38)

where $A_3=-X{w}_{24}^2{w}_{t2}^2+((-2X{w}_{14}{w}_{t1}+T+X)w_{t2}+b_2)w_{24}$, $A_4=-X{w}_{14}^2{w}_{t1}^2+((T+X)w_{t1}+b_1)w_{14}-X$, $X=x^{\beta }/\beta$, and $T=t^{\alpha }/\alpha$.

By substituting the initial condition into Equation (38), we obtain the solution in the following form:

$$u(x,t)={\displaystyle \frac{w_{14}{\mathrm{e}}^{A_5+b_3}+w_{24}{\mathrm{e}}^{A_5+b_4}}{w_{14}{\mathrm{e}}^{A_6+b_3}+w_{24}{\mathrm{e}}^{A_6+b_4}}},$$

(39)

where $A_5=(-2-w_{14}^2{w}_{t1}^2+w_{t1}(-2w_{24}{w}_{t2}+1)w_{14}-w_{24}^2{w}_{t2}^2+w_{24}{w}_{t2})X+b_1{w}_{14}+b_2{w}_{24}$, $A_6=-X{w}_{24}^2{w}_{t2}^2+((-2X{w}_{14}{w}_{t1}+T+X)w_{t2}+b_2)w_{24}-X{w}_{14}^2{w}_{t1}^2+((T+X)w_{t1}+b_1)w_{14}-X$, $X=x^{\beta }/\beta$, and $T=t^{\alpha }/\alpha$.

With the coefficients configuration $\{b_1=1,b_2=1,b_3=1,b_4=1,w_{14}=1,w_{24}=1,w_{t1}=0.5,w_{t2}=0.5,\alpha =0.4,\beta =0.4\}$ specified in Equation (39), the various figures of the Equation (29) are collectively illustrated in Figure 8. The exact solution of the fractional telegraph equation exhibits a smooth and monotonically decreasing behavior as the absolute values of x and t increase. The 3D plot, x-curves, contour plot, and density plot all consistently show this trend, with the solution starting higher at $x=0$ and $t=0$ and gradually declining as x and t change. The visualizations provide a clear understanding of the spatio-temporal dynamics of the solution.

3.3. Fractional Sharma–Tasso–Olever Equation

Let us examine the fractional-order Sharma–Tasso–Olever equation expressed as

$$D_t^{\alpha }u+3a{\left(D_x^{\beta }u\right)}^2+3a{u}^2{D}_x^{\beta }u+3au{D}_x^{2\beta }u+a{D}_x^{3\beta }u=0,$$

(40)

where a is an arbitrary constant, $0<\alpha \le 1$, $0<\beta \le 1$.

Through the variable transformation Equation (3), Equation (40) is rewritten as

$$u_T+3a{u_X}^2+3a{u}^2{u}_X+3au{u}_{XX}+a{u}_{XXX}=0,$$

(41)

Case I: The NNs architecture depicted in Figure 9a, featuring two hidden layers with two neurons each, is employed to solve the governing equation. The first hidden layer employs distinct activation functions, with $tanh(·)$ for its first neuron and $coth(·)$ for the second, while all neurons in the second hidden layer utilize identity mapping $(·)$. According to the analytical formulation Equation (13), we can derive the trial function

$$\begin{array}{cc}\hfill u(X,T)=& \left(w_{13}tanh\left(T{w}_{t1}+X{w}_{x1}+b_1\right)+w_{23}coth\left(T{w}_{t2}+X{w}_{x2}+b_2\right)+b_3\right)w_{3u}\hfill \\ \hfill & +\left(w_{14}tanh\left(T{w}_{t1}+X{w}_{x1}+b_1\right)+w_{24}coth\left(T{w}_{t2}+X{w}_{x2}+b_2\right)+b_4\right)w_{4u}\hfill \\ \hfill & +b_5,\hfill \end{array}$$

(42)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

Substituting Equation (42) into Equation (41) and grouping similar terms involving $\{X,T,tanh(T{w}_{t1}+X{w}_{x1}+b_1),coth(T{w}_{t2}+X{w}_{x2}+b_2)\}$. When all coefficients are set to zero, the system yields 19 constraint groups, mapping to 18 weighted solutions of the equation with bias terms. We give an example of the constraint conditions corresponding to a fully connected NN structure, as shown below:

$$\begin{array}{c}\left\{a=a,b_3=b_3,b_4=b_4,b_5=-b_3{w}_{3u}-b_4{w}_{4u},w_{13}=w_{13},w_{14}=w_{14},w_{23}=w_{23},\right.\hfill \\ w_{24}=w_{24},w_{3u}=w_{3u},w_{4u}=w_{4u},w_{t1}=-a(w_{13}{w}_{3u}+w_{14}{w}_{4u})((w_{13}^2+3w_{23}^2)w_{3u}^2\hfill \\ +2w_{4u}(w_{13}{w}_{14}+3w_{23}{w}_{24})w_{3u}+w_{4u}^2(w_{14}^2+3w_{24}^2)),w_{t2}=-3a(w_{23}{w}_{3u}+w_{24}{w}_{4u})\hfill \\ \times ((w_{13}^2+w_{23}^2/3)w_{3u}^2+2(w_{13}{w}_{14}+w_{23}{w}_{24}/3)w_{4u}{w}_{3u}+w_{4u}^2(w_{14}^2+w_{24}^2/3)),\hfill \\ \left.w_{x1}=w_{13}{w}_{3u}+w_{14}{w}_{4u},w_{x2}=w_{23}{w}_{3u}+w_{24}{w}_{4u}\right\},\hfill \end{array}$$

(43)

and the exact solution satisfying this constraint condition can be expressed as

$$\begin{array}{c}\hfill u(x,t)=tanh\left(T{w}_{t1}+X{w}_{x1}+b_1\right)w_{x1}+coth\left(T{w}_{t2}+X{w}_{x2}+b_2\right)w_{x2},\end{array}$$

(44)

where $w_{x1}$, $w_{x2}$, $w_{t1}$, and $w_{t2}$ are denoted by Equation (43), $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

If one chooses coefficients $\{a=0.5,b_1=3,b_2=3,w_{13}=2,w_{14}=0.5,w_{23}=0.5,w_{24}=0.5,w_{3u}=0.5,w_{4u}=1,\alpha =0.5,\beta =0.5\}$ in the Equation (44), the results are shown in Figure 10. In this way, the solution’s performance becomes more apparent. Figure 10a presents a 3D plot illustrating the solution over the spatio-temporal domain $[0,100]\times [0,20]$. To observe the local behavior of the solution, we plot the x-curves in Figure 10b at interval $t\in [0,20]$. And the contour and density plots of the Equation (40) are shown in Figure 10c and Figure 10d, respectively. From the figure, it can be observed that the solution exhibits a series of fine oscillations, which emerge abruptly on a smooth plane, demonstrating distinct soliton characteristics. The oscillations are unevenly distributed across the plane, with larger amplitudes in certain regions and smaller amplitudes in others. The frequency and amplitude of the oscillations vary both spatially and temporally. The density plot clearly illustrates the vibrational patterns and peaks of the waveform.

Case II: Furthermore, the NNs architecture 6 as shown in Figure 9b is employed to analytically solve the Equation (40). We configure tan(·) activation for all first hidden layer neurons, while implementing identity mapping $(·)$ across the second hidden layer. According to the corresponding analytical formulation Equation (13), we derive

$$\begin{array}{cc}\hfill u(X,T)=& \left(w_{13}tan\left(T{w}_{t1}+X{w}_{x1}+b_1\right)+w_{23}tan\left(T{w}_{t2}+X{w}_{x2}+b_2\right)+b_3\right)w_{3u}\hfill \\ \hfill & +\left(w_{14}tan\left(T{w}_{t1}+X{w}_{x1}+b_1\right)+w_{24}tan\left(T{w}_{t2}+X{w}_{x2}+b_2\right)+b_4\right)w_{4u}\hfill \\ \hfill & +b_5,\hfill \end{array}$$

(45)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

Substituting Equation (45) into Equation (41) and grouping similar terms involving $\{X,T,tan(T{w}_{t1}+X{w}_{x1}+b_1),tan(T{w}_{t2}+X{w}_{x2}+b_2)\}$. Setting every coefficient equal to zero generates 19 distinct constraint groups, each associated with a generalized solution of the equation that includes weights and biases. We give an example of the constraint conditions corresponding to a fully connected NN structure, as shown below:

$$\begin{array}{c}\left\{a=a,b_3=b_3,b_4=b_4,b_5=b_5,w_{13}=-w_{14}{w}_{4u}/w_{3u},w_{14}=w_{14},w_{23}=w_{23},\right.\hfill \\ w_{24}=w_{24},w_{3u}=w_{3u},w_{4u}=w_{4u},w_{t1}=w_{t1},w_{x2}=-w_{23}{w}_{3u}-w_{24}{w}_{4u},\hfill \\ w_{t2}=3a(w_{23}{w}_{3u}+w_{24}{w}_{4u})((b_3^2-w_{23}^2/3)w_{3u}^2+((2b_3{b}_4-\left(2w_{23}{w}_{24}\right)/3)w_{4u}\hfill \\ \left.+2b_3{b}_5)w_{3u}+(b_4^2-w_{24}^2/3)w_{4u}^2+2w_{4u}{b}_4{b}_5+b_5^2),w_{x1}=w_{x1}\right\}.\hfill \end{array}$$

(46)

Consequently, the analytical solution associated with this constraint condition can be expressed as

$$u(x,t)=-tan\left(T{w}_{t2}+X{w}_{x2}+b_2\right)w_{x2}+b_3{w}_{3,u}+b_4{w}_{4,u}+b_5,$$

(47)

where $w_{x2}$ and $w_{t2}$ are denoted by Equation (46), $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

If one chooses coefficients $\{a=1,b_2=1,b_3=1,b_4=1,b_5=1,w_{14}=1,w_{23}=1,w_{24}=1,w_{3u}=1,w_{4u}=1,\alpha =0.5,\beta =0.5\}$ in the Equation (47), the results are shown in Figure 11. In this way, the solution’s performance becomes more apparent. The surface of the image exhibits significant sharp peaks and depressions in certain regions, indicating abrupt changes in the solution within these areas. The distribution of sharp peaks and depressions reveals the nonlinear behavior of the solution in both space and time. The distribution of colors in the density plot exhibits a complex wave-like pattern. This wave-like structure indicates that the solution undergoes periodic or quasi-periodic variations in both space and time, while the changes in color intensity reveal the amplitude and frequency characteristics of the solution.

3.4. Fractional Biological Population Model

We investigate a fractional biological population model in (2 + 1) dimensions with the following formulation:

$$D_t^{\alpha }u=D_x^{2\beta }\left(u^2\right)+D_y^{2\gamma }\left(u^2\right)+h(u^2-r),$$

(48)

where h and r are arbitrary constants, $0<\alpha \le 1$, $0<\beta \le 1$, $0<\gamma \le 1$.

Applying the independent variable transformation

$$X={\displaystyle \frac{x^{\beta }}{\Gamma (1+\beta )}},Y={\displaystyle \frac{y^{\gamma }}{\Gamma (1+\gamma )}},T={\displaystyle \frac{t^{\alpha }}{\Gamma (1+\alpha )}},$$

(49)

we can rewrite Equation (48) as

$$u_T-{\left(u^2\right)}_{XX}-{\left(u^2\right)}_{YY}-h(u^2-r)=0.$$

(50)

Case I: The NNs architecture 7 as shown in Figure 12a is used to solve the equation. The network architecture consists of three inputs, two hidden layers with two neurons each. For the first hidden layer, we adopt the exponential activation function ${\mathrm{e}}^{(·)}$, whereas $(·)$ is implemented throughout the second hidden layer. According to the corresponding analytical formulation, we derive

$$\begin{array}{cc}\hfill u(X,Y,T)=& \left(w_{13}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+Y{w}_{y1}+b_1}+w_{23}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+Y{w}_{y2}+b_2}+b_3\right)w_{3u}\hfill \\ \hfill & +\left(w_{14}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+Y{w}_{y1}+b_1}+w_{24}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+Y{w}_{y2}+b_2}+b_4\right)w_{4u}+b_5,\hfill \end{array}$$

(51)

where $X=x^{\beta }/\beta$, $Y=y^{\gamma }/\gamma$, and $T=t^{\alpha }/\alpha$.

Substituting Equation (51) into Equation (50) and grouping similar terms involving $\{X,Y,T,{\mathrm{e}}^{T{w}_{t1}+X{w}_{x1}+Y{w}_{y1}+b_1},{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+Y{w}_{y2}+b_2}\}$. Setting every coefficient equal to zero generates four distinct constraint groups, each associated with a generalized solution of the equation that includes weights and biases. We give an example of the constraint conditions, as shown below:

$$\begin{array}{c}\left\{h=-4w_{x1}^2-4w_{y1}^2,r={\left(b_3{w}_{3u}+b_4{w}_{4u}+b_5\right)}^2,b_3=b_3,b_4=b_4,b_5=b_5,\right.\hfill \\ w_{13}=w_{13},w_{14}=w_{14},w_{23}=-w_{4u}{w}_{24}/w_{3u},w_{24}=w_{24},w_{3u}=w_{3u},\hfill \\ w_{4u}=w_{4u},w_{t1}=-6\left(w_{x1}^2+w_{y1}^2\right)\left(b_3{w}_{3u}+b_4{w}_{4u}+b_5\right),w_{t2}=w_{t2},\hfill \\ \left.w_{x1}=w_{x1},w_{x2}=w_{x2},w_{y1}=w_{y1},w_{y2}=w_{y2}\right\}.\hfill \end{array}$$

(52)

Based on the aforementioned constraints, we arrive at the following general solution

$$\begin{array}{cc}\hfill u(x,y,t)=& \left(w_{13}{w}_{3u}+w_{14}{w}_{4u}\right){\mathrm{e}}^{-6\left(w_{x1}^2+w_{y1}^2\right)\left(b_3{w}_{3u}+b_4{w}_{4u}+b_5\right)T+X{w}_{x1}+Y{w}_{y1}+b_1}\hfill \\ \hfill & +b_3{w}_{3u}+b_4{w}_{4u}+b_5,\hfill \end{array}$$

(53)

where $X=x^{\beta }/\beta$, $Y=y^{\gamma }/\gamma$, and $T=t^{\alpha }/\alpha$.

By selecting the appropriate values $\{b_1=1,b_3=1,b_4=1,b_5=1,w_{14}=1,w_{13}=1,w_{3u}=1,w_{4u}=1,w_{y1}=0.3,w_{t1}=1,w_{x1}=0.6,w_{x2}=1,\alpha =0.6,\beta =0.6,\gamma =0.6,t=0.5\}$ in Equation (53), the results are shown in Figure 13, which displays the various plots of the Equation (48) over the spatial domain $[0,20]\times [0,20]$. This set of images presents the characteristics of the fractional biological population model from various perspectives, aiding in a more comprehensive understanding of the model’s behavior and evolutionary trends. The figure shows that u increases with the growth of x and y, presenting a smooth, rainbow-like gradient, indicating that the value of u varies continuously in space. From the curve plots, it can be observed that as x and y increase, the curve shifts upward overall and the rate of growth accelerates. From the contour plot, it can be observed that the population density exhibits a gradient trend in both the x and y directions, with the density of the contour lines reflecting the rate of change in population density.

Case II: The NNs architecture 8 as shown in Figure 12b is utilized to solve the equation. For the first hidden layer, we adopt exponential activation functions $tanh(·)$ and ${\mathrm{e}}^{(·)}$, whereas identity mapping $(·)$ and reciprocal $1/(·)$ are implemented throughout the second hidden layer. According to the analytical formulation, the new trial function is

$$\begin{array}{cc}\hfill u(X,Y,T)=& w_{3u}\left(w_{13}tanh\left(T{w}_{t1}+X{w}_{x1}+Y{w}_{y1}+b_1\right)+w_{23}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+Y{w}_{y2}+b_2}+b_3\right)\hfill \\ \hfill & +{\displaystyle \frac{w_{4u}}{w_{14}tanh\left(T{w}_{t1}+X{w}_{x1}+Y{w}_{y1}+b_1\right)+w_{24}{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+Y{w}_{y2}+b_2}+b_4}}\hfill \\ \hfill & +b_5,\hfill \end{array}$$

(54)

where $X=x^{\beta }/\beta$, $Y=y^{\gamma }/\gamma$, and $T=t^{\alpha }/\alpha$.

Substituting Equation (54) into Equation (50) and combining similar terms involving $\{X,Y,T,tanh(T{w}_{t1}+X{w}_{x1}+Y{w}_{y1}+b_1),{\mathrm{e}}^{T{w}_{t2}+X{w}_{x2}+Y{w}_{y2}+b_2}\}$. Setting every coefficient equal to zero generates 10 distinct constraint groups, each associated with a generalized solution of the equation that includes weights and biases. We give an example of the constraint conditions, as shown below:

$$\begin{array}{c}\left\{h=-4w_{x2}^2-4w_{y2}^2,r={\left(\left(b_3{w}_{3u}+b_5\right)w_{14}-w_{4u}/2\right)}^2/w_{14}^2,b_3=b_3,b_4=-w_{14},\right.\hfill \\ b_5=b_5,w_{13}=0,w_{14}=w_{14},w_{23}=w_{23},w_{24}=0,w_{3u}=w_{3u},w_{4u}=w_{4u},\hfill \\ w_{t1}=-3\left(w_{x2}^2+w_{y2}^2\right)\left(\left(b_3{w}_{3u}+b_5\right)w_{14}-w_{4u}/2\right)/w_{14},w_{t2}=2w_{t1},\hfill \\ \left.w_{x1}=w_{x2}/2,w_{x2}=w_{x2},w_{y1}=w_{y2}/2,w_{y2}=w_{y2}\right\}.\hfill \end{array}$$

(55)

Based on the aforementioned constraints, we arrive at the following general solution

$$\begin{array}{cc}\hfill u(x,y,t)=& w_{23}{w}_{3u}{\mathrm{e}}^{T{w}_{t1}+X{w}_{x2}+Y{w}_{y2}+b_2}+b_2+b_3{w}_{23}{w}_{3u}+b_5\hfill \\ \hfill & {\displaystyle \frac{w_{4u}}{w_{14}tanh(2T{w}_{t1}+X{w}_{x2}/2+Y{w}_{y2}/2+b_1)-w_{14}}},\hfill \end{array}$$

(56)

where $w_{t1}$ is denoted by Equation (55), $X=x^{\beta }/\beta$, $Y=y^{\gamma }/\gamma$, and $T=t^{\alpha }/\alpha$.

By selecting the appropriate values $\{b_1=1,b_2=1,b_3=1,b_5=-1,w_{14}=1,w_{23}=1,w_{3u}=-1,w_{4u}=-1,w_{y2}=1,w_{x2}=1,\alpha =1,\beta =1,\gamma =1,t=2\}$ in Equation (56), the results are shown in Figure 14, which displays the various plots of the Equation (48) over the spatial domain $[-5,5]\times [-10,10]$. From the figures, it can be observed that the image of the solution abruptly transitions from a smooth plane to a surface characterized by a series of sharp peaks and depressions. The diagonal structures in the Figure 14d,e suggest that the waveform evolves steadily with spatial progression.

Remark 1. 

To validate the correctness of the derived solutions, they are substituted back into the original equations, yielding exact agreement on both sides. Furthermore, all equation solutions computed by the proposed analytical method are verified using Maple.

Remark 2. 

Several tailored NNs configurations are presented for obtaining analytical solutions to fPDEs, including the fractional wave equation, fractional telegraph equation, fractional Sharma–Tasso–Olever equation, and fractional biological population model. Owing to its adaptable framework, the approach permits alternative network designs to solve these and extended classes of fPDEs.

4. Discussions

In recent years, deep learning approaches have emerged as promising and popular methods for solving various numerical problems of partial differential equations. An NN serves as a computational surrogate to establish the mapping from input variables $(x,t)$ to the output solution $u(x,t)$. The fundamental innovation of PINNs resides in embedding the residual terms of governing equations within the NN’s loss function during optimization. However, PINNs can only provide numerical solutions for solving the equation. Although PINNs have achieved great success in solving equations, there are still many deficiencies. PINNs are data-driven models, thus requiring a large amount of data for training. Optimizing NN parameters necessitates employing iterative algorithms for progressive refinement. Therefore, solving equations using PINNs entails significant computational costs and involves approximation errors.

Owing to their exceptional function approximation properties, NNs enable the derivation of analytical solutions for fPDEs. This work specifically leverages mathematical architectures of NNs to formulate closed-form expressions that solve these equations. Since shallow NNs also possess strong function approximation capabilities, this significantly reduces the complexity of the model. Through judicious configuration of activation functions, layer depth, and neuron density, customized trial solutions can be formulated for diverse fPDE classes. Moreover, the present methodology selectively adopts only the structural framework of neural networks, explicitly excluding their parameter optimization processes. The proposed method utilizes trial functions constructed using NNs to convert the calculation of fPDEs into the computation of a set of nonlinear algebraic equations containing weights and biases. Modifications to the governing equation’s boundary or initial conditions necessitate that PINNs repeat their training procedure. The process requires reformulating the objective function to include the updated physical relationships and then performing full network retraining. Thus, the computational load is heavy and the efficiency is low. Once the general solution to the equation is obtained using the analytical solver, only the corresponding different conditions need to be substituted into the general solution for computation, without the need for repeated calculations.

To validate the computational efficiency of the analytical method in this research, a comparison of numerical examples will be conducted between PINNs and our proposed method.

4.1. Neural Networks-Based Analytical Solver

The proposed methodology is employed to obtain exact solutions for the fractional wave equation under varying initial and boundary conditions, yielding multiple analytical solutions to the Equation (10). The proposed method is implemented using Maple 2024.0 for computation.

Case I: Consider Equation (10) with initial condition Equation (11). Since the general solution to the equation has already been obtained in Section 3.1, the general solution requires substitution of the specified initial and boundary conditions to obtain particular solutions. Substitute Equation (11) into a general solution Equation (19) of Equation (10) to obtain a particular solution.

$$\begin{array}{cc}\hfill u(x,t)=& w_{4u}{\left(w_{24}{\mathrm{e}}^{Tk{w}_{x2}+X{w}_{x2}+b_2}+b_4\right)}^2-w_{24}^2{w}_{4u}{\left({\mathrm{e}}^{X{w}_{x2}+b_2}\right)}^2-2b_4{w}_{24}{w}_{4u}{\mathrm{e}}^{X{w}_{x2}+b_2}\hfill \\ \hfill & -b_4^2{w}_{4u}+2X^2+X,\hfill \end{array}$$

(57)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

For enhanced solution characterization, we adopt the parameter configuration $\{k=1,b_2=1,b_4=1,w_{24}=0.5,w_{4u}=0.4,w_{x2}=0.4,\alpha =1,\beta =1\}$ specified in Equation (57). The solution’s spatio-temporal behavior is visualized through multiple representations in Figure 15, including three-dimensional surface visualization, spatial profile analysis, contour mapping, and density distribution.

Case II: The Equation (10) is investigated under the following initial and boundary conditions:

$$\left\{\begin{array}{c}u(x,0)=x(x-1),\hfill \\ u(0,t)=u(1,t)=0.\hfill \end{array}\right.$$

(58)

Similarly, substitute the Equation (58) into Equation (19) to obtain the solution

$$\begin{array}{c}u(x,t)=X(X-1)\Omega ,\\ \Omega ={\displaystyle \frac{\left(-{\mathrm{e}}^{\left(2Tk+X\right)w_{x2}+2b_2}+{\mathrm{e}}^{\left(2Tk+1\right)w_{x2}+2b_2}+{\mathrm{e}}^{\left(2Tk+2X\right)w_{x2}+2b_2}-{\mathrm{e}}^{\left(2Tk+X+1\right)w_{x2}+2b_2}\right)}{{\mathrm{e}}^{2b_2+w_{x2}}-{\mathrm{e}}^{X{w}_{x2}+2b_2+w_{x2}}+{\mathrm{e}}^{2X{w}_{x,2}+2b_2}-{\mathrm{e}}^{X{w}_{x2}+2b_2}}},\end{array}$$

(59)

where $X=x^{\beta }/\beta$ and $T=t^{\alpha }/\alpha$.

For enhanced solution characterization, we adopt the parameter configuration $\{k=1,b_2=1,w_{4u}=0.45,w_{x2}=0.5,\alpha =1,\beta =1\}$ specified in Equation (59). The solution’s spatio-temporal behavior is visualized through multiple representations in Figure 16, including three-dimensional surface visualization, spatial profile analysis, contour mapping, and density distribution.

4.2. Physics-Informed Neural Networks

We employ PINNs to numerically solve both Case I and Case II presented in Section 4.1, subsequently conducting a systematic comparison between the computational performance and solution accuracy of PINNs and our proposed methodology. The spatio-temporal domain is $[0,1]\times [0,1]$ and $k=\alpha =\beta =1$ in Case I and Case II. The physics-informed loss function in PINNs depends on the governing equation’s initial or boundary conditions. Consequently, varying conditions between Case I and Case II require distinct loss formulations, mandating separate network training procedures for each scenario. We use the following form of $L^2$ relative error

$${\displaystyle \frac{{\left\{{\sum }_i{[u(x_{test,i},t_{test,i})-\tilde{u}(x_{test,i},t_{test,i})]}^2\right\}}^{\frac{1}{2}}}{{\left\{{\sum }_i{\left[u(x_{test,i},t_{test,i})\right]}^2\right\}}^{\frac{1}{2}}}},$$

(60)

mean absolute error, and maximum absolute error to measure the performance of the NNs, where the neural approximation $\tilde{u}$ and analytical solution u are evaluated at discrete test locations $(x_{test,i},t_{test,i})$ for each sample point index i.

The computational framework is implemented in Python 3.9, leveraging TensorFlow’s built-in automatic differentiation functionality. For optimization, we employ the Adam stochastic gradient descent algorithm to minimize the loss function. We initialize the NN parameters using normalized Glorot initialization. For these computational tasks, we implement an NN architecture consisting of 4 hidden layers with a uniform width of 50 neurons per layer. Some other hyperparameters of the NNs, such as the learning rate, the activation function, and the training epoch, are set to $1\times {10}^{-3}$, $tanh\left(x\right)$, and 20,000, respectively. A total of 1000 points are sampled within the spatio-temporal domain, with 500 points sampled on both the initial and boundary conditions.

The performance comparison results between PINNs and our analytical solution method are summarized in

Table 2. Computational performance comparison between the two approaches.

	Computational Performance Indicators	PINNs	Our Method
Case I	Calculation time (s)	372.5405 + 0.0210	0 + 1.03
	Mean absolute error	0.0004140	0
	Maximum absolute error	0.0007336	0
	$L^2$ relative error	0.0001568	0
Case II	Calculation time (s)	456.6720 + 0.0240	0 + 1.17
	Mean absolute error	0.000120	0
	Maximum absolute error	0.0002219	0
	$L^2$ relative error	0.0004117	0

. Since PINNs require an iterative training mechanism to optimize the loss function, the training time is relatively long, and the values 0.0210 and 0.0240 in Table 2 represent the inference time of the NNs. PINNs can obtain approximate solutions to the equations with a certain error, as illustrated in Figure 17 and Figure 18.

Therefore, compared with PINNs, this symbolic computation method can obtain exact solutions for equations without data samples, and it reduces computational costs. To facilitate comparison,

Table 3. Comparison between two approaches.

	Our Method	PINNs
Foundation	Built upon strict mathematical derivations	Integrates physical laws with deep learning
Result type	Analytical solution	Approximate solution
Data dependency	No training data required	Extensive datasets
Model transparency	Clear and logical explanations	Limited interpretability
Computational process	Direct computation without iterations	Requires repeated parameter updates
Calculation cost	Low	High
Precision	Completely accurate	With errors
Optimization algorithm	None	Need
Modify conditions	No need to retrain	Retraining the NNs
Randomness	No	Yes

summarizes the performance metrics of both methodologies. The proposed framework presents an innovative technique for deriving exact solutions to fPDEs, offering significant potential for advancing computational mathematics through its elimination of extensive data requirements and iterative optimization procedures.

5. Conclusions

In this paper, an artificial neural networks-based analytical solver that combines NNs with a symbolic computation method is proposed to analytically solve fPDEs. The computational capability of this analytical approach is verified through successful solutions to the fractional wave equation, fractional telegraph equation, fractional Sharma–Tasso—Olever equation, and fractional biological population model. Employing NN architecture, the solver establishes analytical solutions for fPDEs. New trial formulations emerge from the flexible implementation of different activation functions across multiple network models. This study establishes that 3D surface plots, contour diagrams, and density heatmaps collectively provide a robust framework for analyzing solution dynamics. This research introduces an innovative solution strategy for fPDEs with significant potential for cross-disciplinary implementation in both fundamental and applied research domains.

Compared with traditional numerical methods, the method completely avoids approximation errors and significantly improves the computational efficiency of fPDEs. In contrast to the existing bilinear neural network method [41], our methodology removes the requirement for bilinear transformation, significantly improving accessibility for research workers. Furthermore, not all equations possess a bilinear form. The neural network architecture in our proposed framework exhibits significant adaptability, enabling application to diverse fractional PDEs through customizable configurations of network depth, node density, and nonlinear activation selection. For the first time, NNs are employed to derive exact solutions for fPDEs. However, as an exact solution methodology, the proposed technique is specifically designed for obtaining analytical solutions to fPDEs, rather than numerical approximations. The method employs some basic functions to construct potential analytical solutions for fPDEs, and the activation function provides the basic function for this method. For equations without initial and boundary conditions, the selection of activation functions becomes blind. As the complexity of differential equations and NN models increases, it may lead to higher computational complexity. In the future, we will further apply this method to more complex equations, including high-dimensional fPDEs and nonlinear coupled systems, and attempt to construct new activation functions to reduce computational parameters.