Novel (G ' / G) neural network analysis solver

Abstract

The invention relates to the field of nonlinear partial differential equation modeling and solving, and discloses a novel (G ' / G) neural network analysis solver, a solving method of the solver is based on a neural network architecture, a neural network model comprising an input layer, a hidden layer and an output layer is constructed, and the output of a neural network is used as the solution of a nonlinear partial differential equation; specifically, a neural network model is substituted into a nonlinear partial differential equation, a nonlinear equation about the weight, deviation and activation function of the neural network is generated, an undetermined coefficient method is used for expanding the equation into an algebraic equation set, neural network parameters are determined through a symbol solving method, and finally an accurate solution meeting the nonlinear partial differential equation is generated. According to the method, by combining the nonlinear expression ability and analytical solution of the neural network, discretization errors of a traditional numerical method are avoided, and the efficiency and precision of nonlinear partial differential equation solution are improved.

Description

Technical Field

[0001] The invention relates to the technical field of nonlinear partial differential equation modeling and solution, in particular to a novel (G' / G) neural network analytical solver. Background Art

[0002] Nonlinear partial differential equations (NLPDEs) are important tools for capturing the dynamic behavior of complex systems, including physics, biology, chemistry, and mechanics. Finding analytical solutions to NLPDEs plays a vital role in the study of these physical phenomena and has increasingly become the focus of many scholars.

[0003] However, the solutions of nonlinear partial differential equations are very complex and therefore difficult to solve. Traditional solutions include finite difference method, finite element method, spectral method and imaginary element method, etc. However, these numerical algorithms require 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 (x, t) as input and outputs the solution u(x, t). 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. 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 optimization algorithms. During network training, neural networks are easily trapped in local optimal values ​​and cannot find global optimal values. Therefore, how to improve the training efficiency of the network and develop better optimization algorithms are still hot topics of research.

[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 nonlinear partial differential equations. Therefore, the exact analytical algorithm based on conventional symbolic reasoning cannot be well used to solve nonlinear partial differential equations. Summary of the invention

[0006] In view of the deficiencies of the prior art, the present invention provides a novel (G' / G) neural network analytical solver, which constructs a trial function of the equation by means of a neural network architecture, and then obtains an analytical solution to the nonlinear partial differential equation.

[0007] To achieve the above objectives, the present invention is implemented by the following technical solutions: a novel (G' / G) neural network analytical solution method, which directly establishes the analytical solution of nonlinear partial differential equations based on the neural network architecture, comprises the following steps:

[0008] Construct a nonlinear partial differential equation model to describe the problem to be solved;

[0009] Construct a neural network model including input layer, hidden layer and output layer, and use the output of the neural network as the trial function;

[0010] Substitute the neural network model into the nonlinear partial differential equation to obtain the nonlinear equation about weight, bias and activation function;

[0011] The nonlinear equations are processed using the method of undetermined coefficients to obtain a set of nonlinear algebraic equations about the weights and biases of the neural network;

[0012] Solve the nonlinear algebraic equations to obtain the weights and bias parameters of the neural network model;

[0013] Substitute the obtained weights and biases into the neural network model to obtain the exact solution of the nonlinear partial differential equation.

[0014] Preferably, the nonlinear partial differential equation model is in the following form:

[0015] L{u(x,t)}+N{u(x,t)}=0

[0016] Among them, u(x,t) is the analytical solution of the equation, L{u(x,t)} is a linear operator containing u(x,t) and its partial derivatives, and N{u(x,t)} is a nonlinear operator containing u(x,t) and its partial derivatives.

[0017] Preferably, in the neural network model,

[0018] The neural network input layer takes x and t as input variables;

[0019] The hidden layer consists of at least two layers, each containing multiple neurons, and the neurons are connected through weights and biases;

[0020] The output layer outputs u(x,t) as a trial solution to the nonlinear partial differential equation.

[0021] Preferably, the activation function of the hidden layer is designed based on the following second-order linear ordinary differential equation:

[0022] G"+λG'+μG=0

[0023] Among them, G is the basis function of the activation function, G' and G" are G with respect to its input variable N iare the first and second derivatives of , λ and μ are constant parameters.

[0024] Preferably, the activation function is in the form of the characteristic root discriminant Δ=λ of the equation 2 -4μ value, the activation function is divided into the following three cases:

[0025] When Δ>0, the activation function is based on a hyperbolic function;

[0026] When Δ=0, the activation function is based on a rational function;

[0027] When Δ<0, the activation function is based on a trigonometric function.

[0028] Preferably, the construction of the nonlinear equation comprises the following steps:

[0029] Substitute the neural network's trial function u(x,t) into the nonlinear partial differential equation;

[0030] By using the weights w constructed by the neural network i , Deviation b i And the activation function f i Establish nonlinear relationships;

[0031] Combine like terms and extract the coefficients of each term with respect to the input variables x and t.

[0032] Preferably, the processing of nonlinear equations using the method of undetermined coefficients comprises the following steps:

[0033] After expanding the nonlinear equation, the coefficients of each order term are expressed as i and b i The algebraic expression of

[0034] Setting the coefficients to zero results in a set of nonlinear algebraic equations for the weights and biases of the neural network.

[0035] Preferably, the step of solving the nonlinear algebraic equations is solved by the following method:

[0036] Symbolic methods, based on symbolic computations using computer algebra systems, obtain analytical solutions.

[0037] The present invention also provides a novel (G' / G) neural network analytical solver, comprising:

[0038] An input processing module, for receiving an input of a nonlinear partial differential equation;

[0039] Neural network module, including network parameters and activation functions of input layer, hidden layer and output layer;

[0040] The equation calculation module is used to substitute the neural network trial solution u(x,t) into the nonlinear partial differential equation to generate the network weight w i , Deviation b i and nonlinear equations with input variables;

[0041] The undetermined coefficient processing module is used to merge similar terms of the equation and extract coefficients to construct a i and deviation b i Nonlinear algebraic equations of ;

[0042] Parameter solving module, used to solve nonlinear algebraic equations and obtain the weight parameter w of the neural network i and the deviation parameter b i ;

[0043] The solution module is used to substitute the weight parameters and bias parameters obtained by the solution into the neural network model to obtain the exact solution u(x,t) of the nonlinear partial differential equation.

[0044] The present invention provides a novel (G' / G) neural network analytical solver. It has the following beneficial effects:

[0045] 1. The present invention directly establishes an analytical solution to the nonlinear partial differential equation based on a neural network architecture, which completely avoids approximation errors compared to traditional numerical methods. At the same time, since the numerical discretization process is avoided in the analytical solution process, the computational efficiency of the nonlinear partial differential equation is greatly improved.

[0046] 2. The present invention utilizes a neural network to make the design of the trial function more standardized and clear. The designed neural network architecture is flexible and customizable. By adjusting the number of layers, number of neurons and activation function of the neural network, it can be widely applied to various forms of nonlinear partial differential equations. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a schematic diagram of the solution process of the present invention;

[0048] Figure 2 A neural network image selected for solving a nonlinear partial differential equation model of the present invention;

[0049] Figure 3 This is a flowchart of the neural network parsing solver of the present invention. DETAILED DESCRIPTION

[0050] 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.

[0051] Please see attached Figure 1 -Attached Figure 2 The present invention provides a novel (G' / G) neural network analytical solution method, which directly solves the analytical solution of nonlinear partial differential equations through a neural network architecture, avoiding the discretization error of traditional numerical methods and improving the efficiency and accuracy of analytical solutions. The core idea is to construct a neural network trial function to transform the analytical solution of nonlinear partial differential equations into a nonlinear algebraic equation system solution problem about neural network weights and biases.

[0052] like Figure 1 As shown, the novel (G' / G) neural network analytical solution method may include the following steps:

[0053] S1. Construct nonlinear partial differential equation model;

[0054] S2, building a neural network model, using the output of the neural network as a trial function;

[0055] S3, substituting the neural network model into the nonlinear partial differential equation to obtain a nonlinear equation;

[0056] S4, using the method of undetermined coefficients to process the nonlinear equations and obtain a set of nonlinear algebraic equations about the weights and biases of the neural network;

[0057] S5, solving the nonlinear algebraic equations to obtain the weight and bias parameters of the neural network model;

[0058] S6. Substitute the obtained weights and biases into the neural network model to obtain an analytical solution.

[0059] For step S1, the nonlinear partial differential equation model of the present invention is used to describe the nonlinear partial differential problem to be solved. Specifically, the present invention models the partial differential equation containing linear operators and nonlinear operators to clearly describe the mathematical form of the problem, which is convenient for subsequent analytical solution using neural networks.

[0060] As a possible implementation method, the constructed nonlinear partial differential equation model has the following mathematical form:

[0061] L{u(x,t)}+N{u(x,t)}=0

[0062] in:

[0063] u(x,t) is the analytical solution of the nonlinear partial differential equation to be solved;

[0064] L{u(x,t)} is a linear operator, specifically including u(x,t) and its various partial derivatives with respect to the independent variables x and t;

[0065] N{u(x,t)} is a nonlinear operator, which contains nonlinear terms of u(x,t) and its partial derivatives.

[0066] It should be noted that the specific form of the above model is set according to the needs of the actual problem, and can be either a common single equation model or a coupled equation model. As an option, the independent variables x and t involved in the equation can represent space coordinates and time coordinates, respectively, or other suitable physical quantities.

[0067] In some embodiments, the specific form of the linear operator L{u(x,t)} includes but is not limited to:

[0068] First-order partial derivative operators, for example:

[0069]

[0070] Used to describe the change of the solution u(x,t) in the spatial dimension;

[0071] Second-order partial derivative operators, for example:

[0072]

[0073] Used to describe the diffusive or oscillatory behavior of a solution.

[0074] Specifically, the form of the nonlinear operator N{u(x,t)} can be set according to the characteristics of the physical problem.

[0075] For example:

[0076] In some embodiments, the nonlinear operator may be a product term of a solution to an equation, for example:

[0077]

[0078] Represents the nonlinear advection behavior of the solution;

[0079] In another possible implementation, the nonlinear operator may be a high-order nonlinear term, for example:

[0080] N 2 {u(x,t)}=u(x,t) 3

[0081] Used to describe nonlinear growth or saturation behavior of a solution.

[0082] As an exemplary implementation, the nonlinear partial differential equation model constructed by the present invention can be applied to the following types of problems:

[0083] Fluid dynamics problems, such as the Navier-Stokes equations, are modeled in the form of:

[0084]

[0085] Among them, u is the velocity field, v is the viscosity coefficient;

[0086] Heat conduction problems, such as the diffusion equation, have the following model form:

[0087]

[0088] Where u represents the temperature field and D is the diffusion coefficient;

[0089] Quantum mechanics problems, such as the Schrödinger equation, have a model form of:

[0090]

[0091] Among them, ψ represents the wave function and V(x) is the potential energy function.

[0092] It is understandable that the nonlinear partial differential equation model of the present invention is not limited to the above specific examples. It should be noted that the specific equation form is set according to the physical background and research objectives of the specific problem.

[0093] In this embodiment, the constructed nonlinear partial differential equation model may also include the setting of initial conditions and boundary conditions. As a possible implementation method:

[0094] The initial conditions are used to define the solution at time t = 0, for example:

[0095] u(x,0)=f(x)

[0096] Among them, f(x) is the known initial distribution function;

[0097] Boundary conditions are used to restrict the behavior of the solution at the boundaries of the space, for example:

[0098] u(0,t)=g(t),u(L,t)=h(t)

[0099] Among them, g(t) and h(t) are known boundary functions, and L is the position of the spatial boundary.

[0100] It should be noted that the setting of initial conditions and boundary conditions is crucial to ensure the uniqueness and stability of the solution. As an option, the boundary conditions can be first-class boundary conditions (Dirichlet boundary conditions), second-class boundary conditions (Neumann boundary conditions) or mixed boundary conditions.

[0101] In summary, this embodiment provides a complete method for constructing a nonlinear partial differential equation model, including the setting of linear operators and nonlinear operators, the definition of independent variables, and the description of initial conditions and boundary conditions.

[0102] For step S2, the neural network model of the present invention takes the exact solution of the nonlinear partial differential equation as the solution target, and maps the input variables x, t to the tentative solution u(x, t) of the equation through the mapping ability of the neural network. The neural network model includes an input layer, a hidden layer and an output layer, and its structural design and functional implementation provide a basis for subsequent analytical solutions.

[0103] In one possible implementation, the overall structure of the neural network is a fully connected feedforward neural network. The network has the following characteristics:

[0104] The input layer receives independent variables x and t as input variables of the network;

[0105] The hidden layer contains at least two layers, each layer contains several neurons, and the neurons are connected by weights w i and deviation b i Interconnected;

[0106] The output layer generates a trial solution u(x,t) to the nonlinear partial differential equation.

[0107] It should be noted that the specific number of layers and neurons in the neural network can be adjusted according to the complexity of the partial differential equation to be solved. As an option, simple partial differential equations can use fewer hidden layers and neurons, while for high-dimensional complex equations, the depth and width of the hidden layers can be increased.

[0108] Specifically, the input layer of the neural network takes (x, t) as input variables. Figure 2 As shown, for example, the mathematical expression of the input layer is:

[0109] N 1 =tw t1 +xw x1 +b 1

[0110] N 2 =tw t2 +xw x2 +b 2

[0111] in:

[0112] w t1 、w t2 、w x1 、w x2 is the weight from the input layer to the hidden layer;

[0113] b 1 、b 2 is the corresponding deviation;

[0114] N 1 and N 2 are the outputs of the two neurons in the first hidden layer.

[0115] In the hidden layer, the output of each neuron is activated by the activation function f i Perform nonlinear mapping. As a possible implementation, the neuron output of the hidden layer can be expressed as:

[0116] N 3 =w 23 f 2 (N 2 )+w 1,3 f 1 (N 1 )+b 3

[0117] N 4 =w 24 f 2 (N 2 )+w 14 f 1 (N 1 )+b 4

[0118] in:

[0119] w 1,3 、w 23 、w 14 、w 24 is the weight from the first hidden layer to the second hidden layer;

[0120] b 3 、b 4 is the bias of the second hidden layer;

[0121] f 1 、f 2 is the activation function of the neurons in the first hidden layer.

[0122] The output layer is the output N of the hidden layer. 3 and N 4 As input, generate the final trial solution u(x,t). Specifically, the mathematical mapping of the output layer is:

[0123] u(x,t)=w 3u f 3 (N 3 )+w 4u f 4 (N 4 )+b 5

[0124] in:

[0125] w 3u 、w 4u is the weight from the second hidden layer to the output layer;

[0126] b 5 is the bias of the output layer;

[0127] f 3 、f 4 is the activation function of the neurons in the second hidden layer.

[0128] It should be noted that, in the present invention, the activation function f of the first hidden layer is 1 and f 2 Designated as Its definition is based on a second-order linear ordinary differential equation of the following form:

[0129] G"+λG'+μG=0

[0130] in:

[0131] G=G(N i ), is the input variable N i Function of

[0132] G' and G" are the G's relation to its input variable N i The first and second derivatives of ;

[0133] λ and μ are unknown parameters, which are determined by the characteristics of the nonlinear partial differential equation.

[0134] It should be noted that the activation function is defined by the following relationship:

[0135]

[0136] There are three specific situations:

[0137] 1. When λ 2 When -4μ>0:

[0138]

[0139] 2. When λ 2 When -4μ=0:

[0140]

[0141] 3. When λ 2 When -4μ<0:

[0142]

[0143] in:

[0144] C 1 and C 2 is an arbitrary constant;

[0145] N i represents the input of the i-th neuron in the hidden layer.

[0146] It can be understood that this activation function form based on the solution of ordinary differential equations enhances the ability of neural networks to approximate the analytical solutions of partial differential equations by introducing parameters λ and μ.

[0147] In one possible implementation, the choice of the activation function form depends on the characteristics of the nonlinear partial differential equation.

[0148] It should be noted that the activation function in the present invention is It is not limited to the above forms and can be further expanded and customized according to specific problems.

[0149] In summary, the present invention introduces a special activation function in the first hidden layer The adaptability and analytical expression capabilities of neural networks to nonlinear partial differential equations have been enhanced, laying the foundation for the subsequent generation of trial solutions.

[0150] For step S3, the core of the present invention is to convert the problem of solving the nonlinear partial differential equation into the problem of optimizing the weights and biases of the neural network by using the trial solution u(x, t) of the neural network. To this end, it is necessary to substitute the mathematical form of the neural network model into the target nonlinear partial differential equation and construct a nonlinear equation based on the input variables x and t.

[0151] It should be noted that the form of the target nonlinear partial differential equation is as described in step S1.

[0152] In this embodiment, the form of the neural network output u(x, t) has been determined by step S2.

[0153] In one possible implementation, by substituting the neural network trial function u(x, t) into the linear and nonlinear operators of the partial differential equation, we can obtain a neural network weight w i , Deviation b iAnd the nonlinear equation of the activation function. Specifically, the calculation of the linear operator L{u(x,t)} and the nonlinear operator N{u(x,t)} involves u(x,t) and its partial derivatives of various orders.

[0154] As an example, the specific calculation of the linear operator L{u(x,t)} may include:

[0155]

[0156] By substituting into the neural network model u(x,t), the above operator can be expanded to be related to the network weight w i and deviation b i Related expressions.

[0157] The specific form of the nonlinear operator N{u(x,t)} can be set according to the problem requirements, such as including a higher-order nonlinear term of the solution or a product term of the solution and its derivative. In one implementation, the nonlinear operator may have the following form:

[0158]

[0159] After substituting the trial function constructed using the neural network, the nonlinear operator can also be transformed into i and b i expression.

[0160] It should be noted that during the substitution process, the activation function f i The first and second derivatives of also need to be calculated. According to the definition of the activation function in step S2 above, its derivative form is closely related to the derivative of the function G. For example, for the activation function Its first-order derivative can be further expressed as:

[0161]

[0162] In the process of substitution, it can be understood that the form of the partial differential equation consists of function terms represented by x and t, and the coefficients of these terms are about the neural network parameters w i and b i The algebraic expression of .

[0163] Alternatively, the equation can be expressed as:

[0164] Eqs(w i ,b i ,f i ,x,t)=L{u(x,t)}+N{u(x,t)}=0

[0165] The formal expansion of this equation contains the mathematical expressions of all parameters and activation functions of the neural network.

[0166] It is understandable that after substituting u(x,t) into the partial differential equation, the generated Eqs(w i ,b i ,f i ,x,t) is a nonlinear algebraic equation, and its specific form depends on the original definition of partial differential equations and the structural design of neural networks.

[0167] In summary, this embodiment successfully constructs a nonlinear equation containing neural network weights, biases, and activation functions by substituting the tentative solution u(x, t) of the neural network into the nonlinear partial differential equation, providing a mathematical basis for subsequent steps.

[0168] For step S4, the core goal of the present invention is to transform the problem of solving nonlinear partial differential equations into the problem of solving neural network parameters (weights and biases) by introducing a neural network model. To this end, it is necessary to process the nonlinear equations, specifically using the method of undetermined coefficients to expand the complex equations into a set of algebraic equations about weights and biases.

[0169] In this embodiment, in order to transform the above nonlinear equation into a group of algebraic equations, the method of undetermined coefficients is used.

[0170] In summary, this embodiment expands the nonlinear equation of the partial differential equation into a set of algebraic equations through the method of undetermined coefficients, and transforms the solution problem into a parameter solution problem about the neural network weights and biases by restricting each coefficient to zero, which provides a clear mathematical basis for the parameter solution in subsequent steps.

[0171] For step S5, in this embodiment, all weights w of the neural network are determined by solving the nonlinear algebraic equations constructed in step S4. i and deviation b i Alternatively, the specific solution method may be selected based on the complexity of the algebraic system of equations and the computational requirements.

[0172] In one possible implementation, when the algebraic equations have a clear analytical form and a small number of variables, they can be solved directly using symbolic computation methods. For example, a computer algebra system (such as Mathematica, Maple, or SymPy) can be used to analytically solve the equations to obtain a closed-form expression for each parameter.

[0173] In summary, this embodiment comprehensively processes the nonlinear algebraic equations constructed in step S4 through analytical solution, and obtains all weights and bias parameters of the neural network model.

[0174] As for step S5, it can be understood that in the method of the present invention, the form of the analytical solution u(x, t) is defined by the output of the neural network model, and its specific expression depends on the values ​​of the neural network weights and biases. These parameters have been obtained in step S5 by solving the nonlinear algebraic equations.

[0175] In this embodiment, the process of generating the analytical solution includes the following main steps:

[0176] In a possible implementation, first, the neural network parameters obtained in step S5 are substituted into the mathematical expression of the neural network model. Exemplarily, the form of the trial function u(x, t) is:

[0177] u(x,t)=w 3u f 3 (N 3 )+w 4u f 4 (N 4 )+b 5

[0178] in:

[0179] w 3u 、w 4u and b 5 is the specific parameter value obtained in step S5;

[0180] f 3 and f 4 is the activation function of the hidden layer, and its specific form is defined according to the previous steps;

[0181] N 3 and N 4 is the input to the hidden layer of the neural network.

[0182] It should be noted that the specific form of the activation function f i (N i ) has been described in detail in step S2.

[0183] After substituting the parameters, the analytical solution u(x,t) becomes an explicit function of the independent variables x and t. As an option, the analytical solution can be further simplified for specific problems. For example, if the objective equation has specific boundary conditions or initial conditions, the expression of the analytical solution can be further adjusted according to these conditions to meet specific constraints.

[0184] Specifically, in some embodiments, the form of the analytical solution may include:

[0185] Polynomial functions, such as:

[0186] u(x,t)=c 1 x 2 +c2 t+c 3

[0187] Among them, c 1 、c 2 、c 3 It is a constant determined by the neural network parameters and the activation function form;

[0188] Periodic functions, such as:

[0189] u(x,t)=Asin(ωt+kx)+Bcos(ωt+kx)

[0190] Among them, A, B, ω, and k are determined by parameters and equation characteristics.

[0191] It is understandable that the analytical solution generated by the present invention is not limited to the above form, and its specific expression depends on the structure of the nonlinear partial differential equation and the model design of the neural network.

[0192] In a possible implementation, to verify the correctness of the generated analytical solution, u(x, t) can be substituted back into the target partial differential equation to verify whether the residual of the equation approaches zero. The specific verification method may include:

[0193] Substitute the value of u(x,t) into the equation, calculate the absolute value of the residual, and check whether it is within the given accuracy range;

[0194] The partial derivatives of the analytical solution are symbolized to ensure that the symbolic expression of the residual is equal to zero after substitution into the equation.

[0195] It should be noted that the analytical solution generated in this embodiment can be directly used to describe the dynamic behavior of the target nonlinear partial differential equation without further numerical discretization. This feature makes the method of the present invention have significant advantages in dealing with complex physical, engineering or mathematical problems.

[0196] In summary, this embodiment generates an analytical solution to the nonlinear partial differential equation by substituting the weights and biases obtained in step S5 into the neural network model.

[0197] In general, the present invention directly solves the analytical solution of the nonlinear partial differential equation by constructing a trial solution model based on a neural network. Specifically, the method uses the output of the neural network as a trial function of the equation, generates a nonlinear equation by substituting it into the nonlinear partial differential equation, and uses the method of undetermined coefficients to convert the nonlinear equation into a set of nonlinear algebraic equations about the weights and biases of the neural network, and then obtains the neural network parameters through symbolic solution or numerical optimization methods, thereby generating an analytical solution that satisfies the partial differential equation. The present invention avoids the discretization error of traditional numerical methods, and flexibly adapts to a variety of complex nonlinear partial differential equations through the nonlinear mapping ability of neural networks, while improving the accuracy and efficiency of the solution, providing a highly scalable analytical solution framework.

[0198] The novel (G' / G) neural network analytical solver described below and the novel (G' / G) neural network analytical solution method described above can be referred to each other.

[0199] The present invention also provides a novel (G' / G) neural network analytical solver, comprising:

[0200] An input processing module, for receiving an input of a nonlinear partial differential equation;

[0201] A neural network module, including network parameters and activation functions of the input layer, hidden layer, and output layer, wherein the activation function is customized to be in the form of a hyperbolic function, a trigonometric function, or a rational function according to the equation requirements;

[0202] The equation calculation module is used to substitute the neural network trial solution u(x,t) into the nonlinear partial differential equation to generate the network weight w i , Deviation b i and nonlinear equations with input variables;

[0203] The undetermined coefficient processing module is used to merge similar terms of the equation and extract coefficients to construct a i and deviation b i Nonlinear algebraic equations of ;

[0204] Parameter solving module, used to solve nonlinear algebraic equations and obtain the weight parameter w of the neural network i and the deviation parameter b i ;

[0205] The solution module is used to substitute the weight parameters and bias parameters obtained by the solution into the neural network model to obtain the exact solution u(x,t) of the nonlinear partial differential equation.

[0206] 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.

[0207] Example:

[0208] Please see attached Figure 2 -Attached Figure 3 In order to introduce the specific implementation mode of the present invention in detail, this embodiment takes the following form of the Benjamin-Bona-Mahoney-Peregrine-Burger equation (BBMPB equation) as an example:

[0209] u t -u xxt -au xx +bu x +cuu x +du xxx =0 (1)

[0210] Where a is a positive constant, b is a real constant, and c and d are non-zero real numbers.

[0211] Establish a display model based on a neural network architecture. As we all know, a fully connected neural network model can be used to represent a mathematical mapping; select Figure 2 The neural network architecture with two hidden layers and two neurons in each hidden layer is used to represent the neural network corresponding to the solution u(x, t) in equation (1), and the activation function of the neuron in the first hidden layer is the function defined in step S2. The activation functions of the first neuron and the second neuron in the second hidden layer are the identity mapping (·) and ·) 2 , to construct the neural network expression of u(x,t), we can get:

[0212]

[0213] Substituting the expression corresponding to the above neural network model into the BBMPB equation (1), a nonlinear equation can be obtained.

[0214] According to this equation Combine similar terms and extract coefficients, use the undetermined coefficient method to get the coefficients to 0, and get a set of nonlinear algebraic equations. Then solve this set of nonlinear algebraic equations to get the weights and biases of the God General Network:

[0215]

[0216] Substituting it into (2), we can obtain the symbolic analytical solution u(x, t) of the equation:

[0217]

[0218] It can be seen that the analytical solution is given by This is directly related to the selected activation function of the neural network, so the exact solution of BBMPB is represented by hyperbolic functions, trigonometric functions and rational functions.

[0219] 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 (G' / G) neural network analytical solution method, based on a neural network architecture, directly establishes analytical solutions to nonlinear partial differential equations, characterized in that: The following steps are involved: Construct a nonlinear partial differential equation model to describe the problem to be solved; Construct a neural network model including input layer, hidden layer and output layer, and use the output of the neural network as the trial function; Substitute the neural network model into the nonlinear partial differential equation to obtain the nonlinear equation about weight, bias and activation function; The nonlinear equations are processed using the method of undetermined coefficients to obtain a set of nonlinear algebraic equations about the weights and biases of the neural network; Solve the nonlinear algebraic equations to obtain the weights and bias parameters of the neural network model; Substitute the obtained weights and biases into the neural network model to obtain the exact solution of the nonlinear partial differential equation.

2. A novel (G' / G) neural network analytical solution method according to claim 1, characterized in that: The nonlinear partial differential equation model is in the following form: L{u(x,t)}+N{u(x,t)}=0 Among them, u(x,t) is the analytical solution of the equation, L{u(x,t)} is a linear operator containing u(x,t) and its partial derivatives, and N{u(x,t)} is a nonlinear operator containing u(x,t) and its partial derivatives.

3. A novel (G' / G) neural network analytical solution method according to claim 1, characterized in that: In the neural network model, The neural network input layer takes x and t as input variables; The hidden layer consists of at least two layers, each containing multiple neurons, and the neurons are connected through weights and biases; The output layer outputs u(x,t) as a trial solution to the nonlinear partial differential equation.

4. A novel (G' / G) neural network analytical solution method according to claim 3, characterized in that: The activation function of the hidden layer is designed based on the following second-order linear ordinary differential equation: G"+λG'+μG=0 Among them, G is the basis function of the activation function, G' and G" are G with respect to its input variable N i are the first and second derivatives of , λ and μ are constant parameters.

5. A novel (G' / G) neural network analytical solution method according to claim 4, characterized in that: The activation function is in the form of the characteristic root discriminant Δ=λ of the equation 2 -4μ value, the activation function is divided into the following three cases: When Δ>0, the activation function is based on a hyperbolic function; When Δ=0, the activation function is based on a rational function; When Δ<0, the activation function is based on a trigonometric function.

6. A novel (G' / G) neural network analytical solution method according to claim 1, characterized in that: The construction of the nonlinear equation comprises the following steps: Substitute the trial function u(x,t) constructed using the neural network into the nonlinear partial differential equation; Through the weights w of the neural network i , Deviation b i And the activation function f i Establish nonlinear relationships; Merge like terms and extract coefficients of linearly independent terms with respect to the input variables x and t.

7. A novel (G' / G) neural network analytical solution method according to claim 1, characterized in that: The method of undetermined coefficients for processing nonlinear equations comprises the following steps: After expanding the nonlinear equation, the coefficients of each order term are expressed as i and b i The algebraic expression of Setting the coefficients to zero results in a set of nonlinear algebraic equations for the weights and biases of the neural network.

8. A novel (G' / G) neural network analytical solution method according to claim 1, characterized in that: The step of solving the nonlinear algebraic equations is solved by the following method: Symbolic methods, based on symbolic computations using computer algebra systems, yield analytical solutions.

9. A novel (G' / G) neural network analytical solver, characterized in that: include: An input processing module, for receiving an input of a nonlinear partial differential equation; Neural network module, including network parameters and activation functions of input layer, hidden layer and output layer; The equation calculation module is used to substitute the neural network trial solution u(x,t) into the nonlinear partial differential equation to generate the network weight w i , Deviation b i and nonlinear equations with input variables; The undetermined coefficient processing module is used to merge similar terms of the equation and extract coefficients to construct a i and deviation b i Nonlinear algebraic equations of ; Parameter solving module, used to solve nonlinear algebraic equations and obtain the weight parameter w of the neural network i and the deviation parameter b i ; The solution module is used to substitute the weight parameters and bias parameters obtained by the solution into the neural network model to obtain the exact solution u(x,t) of the nonlinear partial differential equation.

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