Novel fractional order variable transformation neural network analysis solver

Abstract

The invention relates to the field of fractional order partial differential equation modeling, and discloses a novel fractional order variable transformation neural network analytic solver, which depends on a neural network architecture, uses a display model of the neural network architecture as a tentative function of a fractional order partial differential equation, transforms an input variable, and obtains a fractional order variable transformation neural network model. According to the method, a fractional order partial differential equation of a Riemann-Liouville derivative with Jumarie correction is converted into an integer order partial differential equation, and due to the fact that a tentative function is composed of weight and deviation terms in a neural network structure, solving of the fractional order partial differential equation can be converted into calculation of the weight and deviation. Firstly, an equation is converted into a non-linear algebraic equation which is easy to solve by utilizing a heuristic function, then corresponding weight and deviation are determined by utilizing an undetermined coefficient method, and the analytic solver can obtain an accurate symbol analytic solution of a fractional order partial differential equation without a data sample.

Description

Technical Field

[0001] The present invention relates to the technical field of fractional-order partial differential equation modeling, and in particular to a novel fractional-order variable transformation neural network analytical solver. Background Art

[0002] In recent years, fractional-order partial differential equations have been successfully applied to many abnormal phenomena and complex systems in the fields of natural science and engineering technology. The advantage of fractional-order partial differential equations is that they can better describe processes with memory and inheritance characteristics, including solute transport in fractures and porous media, sound wave propagation with frequency-dependent dissipation, laminar and turbulent flows, viscoelastic constitutive laws, optimal control, ecosystems, continuous-time random walks in external fields, fractal mobile and immobile solute migration, and power-law attenuation coefficients of biological tissues. Therefore, finding stable and accurate methods to approximate fractional-order partial differential equations has always been the goal of some researchers.

[0003] However, the solution of fractional-order partial differential equations is very complex and therefore difficult to solve. Traditional solution methods include finite difference method, finite element method, spectral method and imaginary element method, etc. However, this type of numerical algorithm requires numerical discretization, huge data requires a lot of time cost, and there are approximation errors.

[0004] With the rise of artificial intelligence, deep learning has been widely used in the field of science and technology. Due to the high expressive power of neural networks in function approximation, it has been theoretically proven that it is feasible to use neural networks to approximate the solutions of partial differential equations. The work of using physical information neural networks (PINNs) to solve partial differential equations has attracted widespread attention from scholars. Physical information neural networks use the output of neural networks as a proxy model of partial differential equations. It approximates the mapping from space-time coordinates to the solution of the equation, that is, the network takes the dependent variable of the equation as input and outputs the solution. Physical information neural networks incorporate the residual information of partial differential equations into the loss function of the neural network and minimize the training relative to the parameters of the neural network to obtain the optimal model. Fractional-order partial differential equations are different from integer-order partial differential equations because the standard chain rule in integer calculus cannot be applied to fractional-order operators. In order to solve this difficulty, fractional-order physical information neural networks (fPINNs) use traditional numerical methods to numerically discretize fractional-order operators. At the same time, automatic differentiation of integer-order operators and numerical discretization of fractional-order operators are used, so that the residual information of fractional-order partial differential equations can be incorporated into the loss function of the neural network for training. However, since numerical discretization is also performed when solving fractional-order partial differential equations, this method also has approximate errors. Fractional-order physical information neural networks are trained by data-driven, so a large number of training points are required for training, resulting in a high time cost. In addition, neural network training is also limited by the optimization algorithm. During the network training process, the neural network is easily trapped in the local optimal value and cannot find the global optimal value. Therefore, how to improve the training efficiency of the network and develop better optimization algorithms are still hot research topics.

[0005] The exact analytical algorithm based on conventional symbolic reasoning cannot ensure the solution of arbitrary constraint problems, and there is currently no universal analytical algorithm for fractional-order partial differential equations. Therefore, the exact analytical algorithm based on conventional symbolic reasoning cannot be used well to solve fractional-order partial differential equations. Summary of the invention

[0006] In view of the deficiencies in the prior art, the present invention provides a novel fractional-order variable transformation neural network analytical solver, which constructs a trial function of the equation with the help of a neural network architecture, and then obtains an analytical solution to the fractional-order partial differential equation.

[0007] To achieve the above objectives, the present invention is implemented by the following technical scheme: a novel fractional-order variable transformation neural network analytical solution method, comprising the following steps:

[0008] Perform variable transformation on the time variable t and the space variable x to obtain transformed variables T and X;

[0009] Establish a neural network model with multiple hidden layers and neurons, which maps the transformed variables (X, T) to the solution u(X, T) of the fractional-order partial differential equation;

[0010] Substitute the output u(X, T) of the neural network model into the original fractional-order partial differential equation to obtain a nonlinear equation about the weights and biases of the neural network;

[0011] Solving the nonlinear equation using the method of undetermined coefficients to determine the weights and biases of the neural network;

[0012] Substitute the obtained weights, biases and variable transformations into the neural network model to obtain the analytical solution u(x, t) of the fractional-order partial differential equation.

[0013] Optionally, the fractional-order partial differential equation is expressed as:

[0014]

[0015] Wherein, u=u(x, t) is the function to be solved; represents the fractional derivative of u with respect to time t with order α; represents the fractional derivative of u with respect to space x with order β; α and β are the orders of the fractional derivative, satisfying 0<α≤1, 0<β≤1; f is a function containing u and its fractional derivatives of various orders.

[0016] Optionally, the fractional derivative and is the Jumarie-modified Riemann-Liouville derivative, defined as:

[0017]

[0018] in, represents the fractional derivative operator with respect to time t; Γ is the gamma function; f(s) is the value of function f at s; f(0) is the value of function f at s = 0; f ( an ) (t) represents the (α-n)th order derivative of the function f(t); [f (α-n) (t)] (n) Indicates f (a-n) (t) Perform n integer-order derivative operations.

[0019] Optionally, the variable transformation includes:

[0020] Fractional variable transformation of time variables:

[0021]

[0022] Where t is the time variable, α is the order of the time fractional derivative, and Γ is the gamma function;

[0023] The fractional variable transformation form of spatial variables is:

[0024]

[0025] Where x is a spatial variable, β is the order of the spatial fractional derivative, and Γ is the gamma function.

[0026] Optionally, after the variable transformation, the fractional derivative is converted into an integer derivative, satisfying:

[0027]

[0028] in, and denote the fractional derivative operators with respect to t and x respectively; is the function after time variable transformation; is the function after spatial variable transformation.

[0029] Optionally, the neural network model is a fully connected feedforward neural network having n hidden layers, each hidden layer having m neurons, and the output of the nth neuron in the first hidden layer is:

[0030] z 1n =f 1n (sum) = f 1n (w xn X+w tn T)+b 1n

[0031] Among them, f 1n represents the activation function of the nth neuron in the first layer, w xn and w tn denote the weights of the nth neuron related to X and T, respectively, and b 1n is the deviation term;

[0032] The output of the nth neuron in the nth layer of the neural network model is:

[0033]

[0034] Among them, f nn represents the activation function of the nth neuron in the nth layer, Z (n-1)1 is the output of the first neuron in the n-1th layer, Z (n-1)m is the output of the mth neuron in the n-1th layer, and is the weight coefficient, bnn is the deviation term;

[0035] The output layer of the neural network model is expressed as:

[0036] u(X,T)=w 1u Z n1 +w 2u Z n2 +…+w mu z nm +b

[0037] Among them, u(X, T) is the output of the neural network model; w 1u , w 2u , ..., W mu is the weight of the output layer, connecting the 1st to the mth neurons of the nth layer to the output layer respectively; Z n1 , z n2 , ..., z nm is the output of the 1st to mth neurons in the nth layer; b is the bias term of the output layer, m is the number of neurons in the nth layer; n is the total number of layers in the neural network.

[0038] Optionally, the nonlinear equation about the neural network weights and biases is expressed as:

[0039] Eqs(w i , b i , f i , X, T) = 0

[0040] Among them, w i is the weight; b i is the deviation; f i is the activation function.

[0041] Optionally, the step of solving the nonlinear equation using the method of undetermined coefficients to determine the weights and biases of the neural network includes:

[0042] In the resulting equation, the terms with the same weights and biases are merged to simplify the form of the equation;

[0043] Extract all coefficients before weights and deviations from the combined equation and set these coefficients as unknowns in the equation;

[0044] Using the extracted coefficients as variables, a nonlinear algebraic equation system is established;

[0045] The set of algebraic equations is solved by setting the extracted coefficients to zero and applying the method of undetermined coefficients to find the specific values ​​of each weight and bias.

[0046] Optionally, the step of substituting the obtained weights, biases and variable transformations into the neural network model to obtain an analytical solution to the fractional-order partial differential equation includes:

[0047] The weight w obtained by the method of undetermined coefficients i and deviation b i Substitute into each neuron of the neural network;

[0048] Using the updated neural network parameters, the output of the neural network is calculated based on the input variables X and T, which represents the approximate solution u(X, T) of the equation;

[0049] Applying the inverse process of the variable transformation defined previously, we transform u(X, T) back to the original variable space and obtain u(x, t), which is the analytical solution to the fractional-order partial differential equation.

[0050] The present invention also provides a novel fractional-order variable transformation neural network analytical solver, comprising:

[0051] A variable transformation module is used to transform the time variable and space variable of the fractional-order partial differential equation into standardized time and space variables respectively;

[0052] A neural network module having at least two hidden layers, each of which includes a plurality of neurons, for receiving the transformed time and space variables and outputting a solution to the equation;

[0053] The solution module solves the nonlinear algebraic equations generated by the neural network model based on the method of undetermined coefficients, obtains the weights and biases of the neural network, and outputs the analytical solution of the fractional-order partial differential equation through the neural network.

[0054] The present invention provides a novel fractional-order variable transformation neural network analytical solver, which has the following beneficial effects:

[0055] 1. By introducing variable transformation, the present invention effectively transforms the complex fractional-order derivative problem into the integer-order derivative problem, thereby greatly simplifying the solution process of the fractional-order partial differential equation. Through this transformation, the computational complexity caused by processing fractional-order derivatives in the traditional method is reduced, making the solution process more efficient and easy to implement.

[0056] 2. The present invention solves the fractional-order partial differential equation analytically by constructing a neural network model, thus avoiding the discretization error commonly seen in traditional numerical discretization methods. The method of undetermined coefficients is used to solve the weights and biases of the neural network, further ensuring the improvement of the solution accuracy. Compared with the numerical approximation method, the analytical solution method can theoretically provide higher accuracy.

[0057] 3. Neural networks have extremely high expressive power when approximating complex functions, which enables the present invention to quickly solve fractional-order partial differential equations by training a small number of neural network parameters. Compared with traditional numerical solution methods, this method reduces the reliance on a large number of data points, thereby significantly shortening the calculation time, and is suitable for engineering and scientific application scenarios that require fast calculations.

[0058] 4. The neural network architecture designed by the present invention has flexible adaptability. By adjusting the number of neural network layers, the number of neurons and the activation function, it can be widely applied to various forms of fractional-order partial differential equations. Whether it is a linear equation or a nonlinear equation, it can be solved by reasonably designing the network architecture, and has a wide range of application potential.

[0059] 5. The present invention can not only solve typical fractional-order partial differential equations, but also flexibly adjust the network structure and solution process according to the specific needs of the problem, and is applicable to problems involving fractional-order dynamics in physics, engineering, control systems, bioengineering, etc. This versatility and extensibility make this method have important application prospects in many fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a schematic diagram of the solution method flow of the solver of the present invention;

[0061] Figure 2 A neural network model according to an embodiment of the present invention;

[0062] Figure 3 A neural network image selected for a dependent variable in a fractional-order partial differential equation model according to an embodiment of the present invention;

[0063] Figure 4 A schematic diagram of a method flow chart of an embodiment of the present invention;

[0064] Figure 5 It is a schematic diagram of the device structure of the present invention.

[0065] Among them, 10, variable transformation module; 20, neural network module; 30, solution module. DETAILED DESCRIPTION

[0066] The following will be combined with the drawings in the specification of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0067] Please see attached Figure 1The present invention provides a novel fractional variable transformation neural network analytical solution method, which aims to solve fractional partial differential equations (FPDEs) efficiently and accurately. The method transforms the fractional derivatives into integer derivatives by variable transformation, and combines the powerful function approximation ability of the neural network to achieve the analytical solution of FPDEs. The specific implementation of the solution method is described in detail below.

[0068] S1. Modeling of fractional-order partial differential equations

[0069] First of all, modeling the fractional-order partial differential equation is the key foundation of the entire solution process. By defining the form of the fractional-order partial differential equation, clarifying the characteristics of the fractional-order derivative and its fractional-order derivative relationship in time and space dimensions, a theoretical basis is provided for the subsequent solution steps. The focus of this step is to lay the foundation for variable transformation and neural network solution methods through the modified Riemann-Liouville derivative formula and related formulas.

[0070] In this embodiment, the fractional-order partial differential equation to be solved is first defined, and the general form of the equation is:

[0071]

[0072] in:

[0073] u=u(x, t) is the function to be solved;

[0074] represents the fractional derivative of u with respect to time t with order α;

[0075] represents the fractional derivative of u with respect to space x with order β;

[0076] α and β are the orders of fractional derivatives, satisfying 0<α≤1, 0<β≤1;

[0077] f is a function containing u and its fractional derivatives.

[0078] This equation form is applicable to many physical and engineering problems with memory or inheritance effects.

[0079] The fractional derivative uses the Jumarie modified Riemann-Liouville derivative, which is defined as:

[0080]

[0081] in:

[0082] represents the fractional derivative operator with respect to time t;

[0083] Γ is the gamma function;

[0084] f(s) is the value of function f at point s;

[0085] f(0) is the value of function f at s=0;

[0086] f (α-n) (t) represents the (α-n)th order derivative of the function f(t);

[0087] [f (α-n) (t)] (n) Indicates f (α-n) (t) Perform n integer-order derivative operations.

[0088] This definition is used to deal with the problem of non-local effects, and the introduction of the gamma function Γ(·) ensures the consistency and standardization of the fractional derivatives.

[0089] In order to further simplify the processing of fractional-order derivatives, this embodiment provides several key formulas, which can effectively convert fractional-order derivative operations into integer-order derivative operations:

[0090] 1. Fractional derivatives of polynomials:

[0091] For a polynomial x m The fractional derivative of is as follows:

[0092]

[0093] This formula converts the operation of fractional derivatives into the form of power functions through the ratio of gamma functions, thereby simplifying the complexity of subsequent variable transformation and derivative solution.

[0094] 2. Fractional derivative of the product:

[0095] For the product of two functions f(t) and g(t), its fractional derivatives follow the following product rule:

[0096]

[0097] This rule is similar to the product rule for integer-order derivatives, and helps to decompose fractional-order derivative operations of complex functions in neural network models.

[0098] 3. Fractional derivatives of composite functions:

[0099]

[0100] This conclusion ensures that after the variable transformation, the fractional-order derivatives of the composite function can be processed correctly, and then the fractional-order derivatives can be approximated by integer-order derivative operations.

[0101] In this embodiment, these formulas provide theoretical support for the subsequent variable transformation (step S2) and neural network solution (step S3). Through these formulas, fractional order derivatives can be effectively converted into more tractable integer order derivatives, laying a solid mathematical foundation for the construction and solution of the neural network model.

[0102] S2. Variable transformation

[0103] By appropriately transforming the input variables of the fractional-order partial differential equation, the originally complex fractional-order derivative operation is transformed into an integer-order derivative operation, thereby simplifying the solution process. This variable transformation not only retains the characteristics of the fractional-order derivative, but also lays the foundation for the construction of the subsequent neural network model.

[0104] In this embodiment, in order to process the fractional-order derivatives in the fractional-order partial differential equation, the time variable t and the space variable x are transformed as follows:

[0105]

[0106] Among them, t is the time variable, α is the order of the time fractional derivative, x is the space variable, β is the order of the spatial fractional derivative, and Γ is the gamma function used to normalize the fractional derivative. This transformation converts the fractional derivatives in time and space into standard integer derivatives, thereby simplifying subsequent calculations.

[0107] Then, based on step S1, the following derivation operation is performed:

[0108]

[0109] Through this variable transformation, the originally complex fractional derivatives are converted into regular integer derivatives, thereby simplifying subsequent calculations. and is normalized to the T and X dimensions and then solved analytically in these new coordinate systems.

[0110] In this embodiment, the introduction of this variable transformation not only helps to simplify the processing of fractional derivatives, but also enables the neural network to more efficiently model and approximate the solution of the equation. This provides a standardized input format for the definition of the input layer of the neural network model in the subsequent steps, and ensures that the neural network model can effectively handle the approximation problem of fractional derivatives. After the variable transformation, the neural network solver can use standard integer derivatives to process complex fractional partial differential equations.

[0111] S3. Build a neural network model

[0112] After completing the variable transformation of the fractional-order partial differential equation, a neural network model is constructed to solve the equation analytically. The neural network model uses the transformed variables as input and approximates the solution of the fractional-order partial differential equation by optimizing its weights and biases.

[0113] In this embodiment, a multi-layer feedforward neural network model is used to approximate the solution of the fractional-order partial differential equation. The input of the neural network is the variables T and X processed in step S2, and the output is the solution u(X, T) of the equation. The structure of the neural network model is:

[0114] 1. Input layer: receives T and X after variable transformation as input, that is and

[0115] 2. Hidden layer: The neural network contains at least two hidden layers, each of which contains several neurons, and different layers use different activation functions. The activation function of the first hidden layer is exp(·), and the activation function of the second hidden layer is 1 / x. Through these activation functions, the network can better capture the characteristics of fractional derivatives.

[0116] The output of the nth neuron in layer 1 is:

[0117] z 1n =f 1n (sum) = f 1n (w xn X+w tn T)+b 1n

[0118] Among them, f 1n represents the activation function of the nth neuron in the first layer, w xn and w tn denote the weights of the nth neuron related to X and T, respectively, and b 1n is the deviation term.

[0119] The output of the nth neuron in the nth hidden layer is:

[0120]

[0121] Among them, f nn represents the activation function of the nth neuron in the nth layer, Z (n-1)1 is the output of the first neuron in the n-1th layer, Z (n-1)m is the output of the mth neuron in the n-1th layer, and is the weight coefficient, b nn is the deviation term.

[0122] 3. Output layer: The final output of the neural network model approximates the solution of the fractional-order partial differential equation. The expression of the output layer is:

[0123] u(X,T)=w 1u z n1 +w 2u z n2 +…+w mu z nm +b

[0124] Among them, u(X, T) is the output of the neural network model; w 1u , w 2u , ..., w mu is the weight of the output layer, connecting the 1st to the mth neurons of the nth layer to the output layer respectively; z n1 , z n2 , ..., z nm is the output of the 1st to mth neurons in the nth layer; b is the bias term of the output layer, m is the number of neurons in the nth layer; n is the total number of layers in the neural network.

[0125] In this embodiment, after the neural network model is constructed, the output of the neural network model u(X, T) is substituted into the fractional order partial differential equation to form the neural network weight w i and deviation b i By optimizing the weights and biases of the neural network and using the method of undetermined coefficients to solve these nonlinear equations, we can get the optimal solution to the approximation equation.

[0126] Ultimately, the neural network model approximates the analytical solution u(X, T) of the equation by adjusting weights and biases, and can flexibly adjust the architecture and activation function of the neural network according to the given fractional-order partial differential equation to ensure adaptability and accuracy to different problems.

[0127] S4. Substitute the original equation and construct a nonlinear equation, and use the undetermined coefficient method to solve the network parameters

[0128] After completing the construction of the neural network model and the transformation of variables, the output of the neural network model is substituted into the fractional-order partial differential equation to form a nonlinear equation about weights and biases. Then, these equations are solved by the method of undetermined coefficients to obtain the optimal weights and biases, and then the analytical solution of the equation is obtained.

[0129] In this embodiment, the output u(X, T) of the neural network model established in step S3 is substituted into the transformed form of the fractional-order partial differential equation. The equation after substitution transforms the weight w in the neural network into i , Deviation b i And the activation function fi is expressed as a parameter in the equation, forming a nonlinear equation of the following form:

[0130] Eqs(w i , b i , f i , X, T) = 0

[0131] Among them, X and T are independent variables, w i and b i are the weights and biases of the neural network, respectively, and f i is the activation function.

[0132] In this embodiment, in order to simplify the solution process, the activation function is first selected, and then the equation is simplified to include only the weight w i , Deviation b i and the form of the independent variables X and T, we get the following equation:

[0133] Eqs(w i , b i , X, T) = 0

[0134] Next, combine the similar terms in the equation and extract all the coefficient terms. Using the undetermined coefficient method, set these coefficient terms to 0 to form a set of weights w i and deviation b i By solving these equations, we can get the optimal weights and biases of neurons in each layer of the neural network model.

[0135] Specifically, the parameters of the network are solved by the following steps:

[0136] 1. Merge like terms: After substituting the neural network into the equation, sort out the resulting equation and merge the same power terms about X and T.

[0137] 2. Extract coefficients: Extract the coefficients of each order of X and T from the equation.

[0138] 3. Undetermined coefficient method: Use the undetermined coefficient method to set all coefficients to 0 and obtain a set of weights w i and deviation b i Nonlinear algebraic equations.

[0139] By solving these equations, the weights and biases of the neurons in each layer are determined.

[0140] S5. Obtain analytical solutions to fractional-order partial differential equations

[0141] After obtaining the optimal weights and biases of the neural network by the method of undetermined coefficients, the final analytical solution can be obtained by substituting these parameters into the neural network model and performing a reverse transformation of the variables. The focus of step S5 is to map the analytical solution from the coordinate system X and T after the variable transformation back to the original time and space variables t and x, thereby obtaining the final solution of the fractional-order partial differential equation.

[0142] In general, the present invention directly establishes an analytical solution to the fractional-order partial differential equation based on the neural network architecture. Compared with the traditional numerical method, it completely avoids the approximation error. At the same time, since the numerical discretization process is avoided in the analytical solution process, the computational efficiency and solution accuracy of the fractional-order partial differential equation are greatly improved.

[0143] In order to better understand the present invention, the above method is described in detail below in conjunction with specific embodiments.

[0144] Example:

[0145] like Figure 4 As shown, this embodiment takes the following form of fractional order telegraph equation as an example:

[0146]

[0147] Before building the neural network model, the input variables of the network are transformed:

[0148]

[0149] Obviously, the above variable transformation ultimately converts the derivation operation of fractional-order derivatives into the derivation operation of integer-order derivatives.

[0150] like Figure 2 As shown, a display model based on a neural network architecture is established. It is well known that a fully connected neural network model can be used to represent a mathematical mapping;

[0151] like Figure 3 As shown, a neural network architecture with two hidden layers and two neurons in each hidden layer is selected to represent the neural network corresponding to the solution u(X,T) in equation (1). The activation function of the neurons in the first hidden layer is exp(·), and the activation function of the neurons in the second layer is 1 / x. To construct the expression of the neural network of u(X,T), we can get:

[0152]

[0153] Substituting the expression corresponding to the above neural network model into the fractional-order telegraph equation (1), we can obtain the following nonlinear equation:

[0154]

[0155] Combine the same terms in this equation and extract the coefficients. Use the undetermined coefficient method to set the coefficients to 0. Then solve this set of nonlinear algebraic equations to obtain the weights and deviations:

[0156]

[0157] Substituting it into (2), we can obtain the symbolic analytical solution u(X,T) of the equation:

[0158]

[0159] Finally and Substituting back into u(X,T) we get:

[0160]

[0161] It can be seen that the analytical solution is composed of exponential functions, which is directly related to the selected activation function of the neural network.

[0162] The novel fractional-order variable transformation neural network analytical solver described below and the novel fractional-order variable transformation neural network analytical solution method described above can be referred to each other.

[0163] Please see attached Figure 5 The present invention also provides a novel fractional-order variable transformation neural network analytical solver, comprising:

[0164] A variable transformation module 10, used for transforming the time variable and the space variable of the fractional-order partial differential equation into standardized time and space variables respectively;

[0165] A neural network module 20, having at least two hidden layers, each including a plurality of neurons, for receiving the transformed time and space variables and outputting a solution to the equation;

[0166] The solution module 30 solves the nonlinear algebraic equation generated by the neural network model based on the method of undetermined coefficients, obtains the weight and bias of the neural network, and outputs the analytical solution of the fractional-order partial differential equation through the neural network.

[0167] The solver of this embodiment can be used to execute the above method embodiment, and its principles and technical effects are similar, which will not be repeated here.

[0168] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A novel fractional-order variable transformation neural network analytical solution method, characterized in that: The following steps are involved: Perform variable transformation on the time variable t and the space variable x to obtain transformed variables T and X; Establish a neural network model with multiple hidden layers and neurons, which maps the transformed variables (X, T) to the solution u(X, T) of the fractional-order partial differential equation; Substitute the output u(X, T) of the neural network model into the original fractional-order partial differential equation to obtain a nonlinear equation about the weights and biases of the neural network; Solving the nonlinear equation using the method of undetermined coefficients to determine the weights and biases of the neural network; Substitute the obtained weights, biases and variable transformations into the neural network model to obtain the analytical solution u(x, t) of the fractional-order partial differential equation.

2. The novel fractional-order variable transformation neural network analytical solution method according to claim 1 is characterized in that: The fractional order partial differential equation is expressed as: Wherein, u=u(x, t) is the function to be solved; represents the fractional derivative of u with respect to time t with order α; represents the fractional derivative of u with respect to space x with order β; α and β are the orders of the fractional derivative, satisfying 0<α≤1, 0<β≤1; f is a function containing u and its fractional derivatives of various orders.

3. The novel fractional-order variable transformation neural network analytical solution method according to claim 2 is characterized in that: The fractional derivative and is the Jumarie-modified Riemann-Liouville derivative, defined as: in, represents the fractional derivative operator with respect to time t; Γ is the gamma function; f(s) is the value of function f at s; f(0) is the value of function f at s = 0; f (a-n) (t) represents the (α-n)th order derivative of the function f(t); [f (α-n) (t)] (n) Indicates f (a -n) (t) Perform n integer-order derivative operations.

4. The novel fractional-order variable transformation neural network analytical solution method according to claim 1 is characterized in that: The variable transformation includes: Fractional variable transformation of time variables: Where t is the time variable, α is the order of the time fractional derivative, and Γ is the gamma function; The fractional variable transformation form of spatial variables is: Where x is a spatial variable, β is the order of the spatial fractional derivative, and Γ is the gamma function.

5. The novel fractional-order variable transformation neural network analytical solution method according to claim 4 is characterized in that: After the variable transformation, the fractional derivative is converted to an integer derivative, satisfying: in, and denote the fractional derivative operators with respect to t and x respectively; is the function after time variable transformation; is the function after spatial variable transformation.

6. According to claim 1, a novel fractional-order variable transformation neural network analytical solution method is characterized in that: The neural network model is a fully connected feedforward neural network with n hidden layers, each hidden layer has m neurons, and the output of the nth neuron in the first hidden layer is: z 1n =f 1n (sum)=f 1n (w xn X+w tn T)+b 1n Among them, f 1n represents the activation function of the nth neuron in the first layer, w xn and w tn denote the weights of the nth neuron related to X and T, respectively, and b 1n is the deviation term; The output of the nth neuron in the nth layer of the neural network model is: Among them, f nn represents the activation function of the nth neuron in the nth layer, z (n-1)1 is the output of the first neuron in the n-1th layer, z (n-1)m is the output of the mth neuron in the n-1th layer, and is the weight coefficient, b nn is the deviation term; The output layer of the neural network model is expressed as: u(X,T)=w 1u z n1 +w 2u z n2 +…+w mu z nm +b Among them, u(X,T) is the output of the neural network model; w 1u ,w 2u ,…,w mu is the weight of the output layer, connecting the 1st to the mth neurons of the nth layer to the output layer respectively; z n1 ,z n2 ,…,z nm is the output of the 1st to mth neurons in the nth layer; b is the bias term of the output layer, m is the number of neurons in the nth layer; n is the total number of layers in the neural network.

7. According to claim 1, a novel fractional-order variable transformation neural network analytical solution method is characterized in that: The nonlinear equation for the neural network weights and biases is expressed as: Eqs(w i ,b i ,f i ,X,T)=0 Among them, w i is the weight; b i is the bias; fi is the activation function.

8. According to claim 1, a novel fractional-order variable transformation neural network analytical solution method is characterized in that: The step of solving the nonlinear equation using the method of undetermined coefficients to determine the weights and biases of the neural network includes: In the resulting equation, the terms with the same weights and biases are merged to simplify the form of the equation; Extract all coefficients before weights and deviations from the combined equation and set these coefficients as unknowns in the equation; Using the extracted coefficients as variables, a nonlinear algebraic equation system is established; The set of algebraic equations is solved by setting the extracted coefficients to zero and applying the method of undetermined coefficients to find the specific values ​​of each weight and bias.

9. The novel fractional-order variable transformation neural network analytical solution method according to claim 1 is characterized in that: The step of substituting the obtained weights, biases and variable transformations into the neural network model to obtain an analytical solution to the fractional-order partial differential equation comprises: The weight w obtained by the method of undetermined coefficients i and deviation b i Substitute into each neuron of the neural network; Using the updated neural network parameters, the output of the neural network is calculated based on the input variables X and T, which represents the approximate solution u(X,T) of the equation; Applying the inverse process of the variable transformation defined previously, we transform u(X,T) back to the original variable space and obtain u(x,t), which is the analytical solution to the fractional-order partial differential equation.

10. A novel fractional-order variable transformation neural network analytical solver, applied to the solution method according to any one of claims 1 to 9, characterized in that: include: A variable transformation module is used to transform the time variable and space variable of the fractional-order partial differential equation into standardized time and space variables respectively; A neural network module having at least two hidden layers, each of which includes a plurality of neurons, for receiving the transformed time and space variables and outputting a solution to the equation; The solution module solves the nonlinear algebraic equations generated by the neural network model based on the method of undetermined coefficients, obtains the weights and biases of the neural network, and outputs the analytical solution of the fractional-order partial differential equation through the neural network.

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