Novel fractional order sub-equation neural network analysis solver

Abstract

The invention relates to the field of fractional calculus and partial differential equation solving, and discloses a novel fractional sub-equation neural network analysis solver, which comprises the following steps: constructing a full-connection neural network model, and substituting the formula into a fractional partial differential equation to obtain a fractional sub-equation neural network analysis solver; and a Riemann-Liouville fractional derivative is corrected by utilizing Jumarie which meets a chain rule, so that the problem is converted into an algebraic equation form. Unknown variables and coefficients of the integer order partial differential equation are extracted by using a symbol calculation tool, an overdetermined nonlinear algebraic equation set is generated, and the weight and bias of the neural network are obtained by solving the equation set. And finally, optimizing the preliminary analytical solution in combination with a special solution of a fractional order Riccati equation, and generating a final analytical solution comprising a generalized fractional order hyperbolic function, a generalized fractional order trigonometric function and a Mittag-Leffler function form. According to the method, the analytic solution efficiency is improved, the expression range of understanding is expanded, and meanwhile, the mathematical characteristics of fractional derivatives are reserved.

Description

Technical Field

[0001] The invention relates to the technical field of fractional-order calculus and partial differential equation solution, and in particular to a novel neural network analytical solver for fractional-order sub-equations. Background Art

[0002] Differential equations have a wide range of applications in many fields, including engineering. It is the backbone of our understanding of how systems evolve over time and provides a mathematical framework for predicting and controlling various phenomena. In current scientific research and engineering practice, we often encounter complex systems with memory and hereditary characteristics, and the behavior of these systems is often difficult to accurately describe using traditional integer-order differential equations. Fractional calculus, as a more general mathematical tool, can more accurately capture the dynamic characteristics of these systems by introducing fractional-order derivatives. However, unlike integer derivatives, fractional-order derivatives are non-local, which means that they depend not only on the value of the current point, but also on the value within the entire domain of definition. This property makes it difficult to solve the analytical solution of fractional-order partial differential equations using traditional calculus methods. To solve this problem, researchers began to explore the use of machine learning methods to solve fPDEs, among which fractional sub-equation neural networks (fSENNs) have attracted much attention due to their unique advantages.

[0003] In order to find analytical solutions to partial differential equations, scientists have conducted many explorations. For example, the extended tanh function method makes full use of the Riccati equation with parameters and uses its solution to replace the tanh function in the tanh function method. Some scientists have generalized the Hirota bilinear operator and made full use of the generalized Hirota bilinear operator to successfully solve a series of nonlinear equations. They can also use the fractional order subequation method to solve the analytical solutions of nonlinear partial differential equations. At the same time, scholars use The expansion method obtains the traveling wave solution of the nonlinear evolution equation. In recent years, some scholars have cleverly combined the generalized Hirota bilinear operator with the neural network to create the bilinear neural network method, which is a powerful tool for solving analytical solutions to partial differential equations.

[0004] Although there are many methods for solving partial differential equations, there are very few methods for fractional partial differential equations. In recent years, some scholars have cleverly combined the generalized Hirota bilinear operator with neural networks to create a bilinear neural network method, which is a powerful tool for solving partial differential equations. Based on this idea, the present invention proposes a new type of neural network analytical solver for fractional sub-equations. Summary of the invention

[0005] In view of the deficiencies in the prior art, the present invention provides a novel fractional-order sub-equation neural network analytical solver, which solves the problem that traditional methods are difficult to efficiently solve analytical solutions to fractional-order partial differential equations.

[0006] To achieve the above objectives, the present invention is implemented by the following technical scheme: a novel fractional order sub-equation neural network analytical solver, comprising the following steps:

[0007] By selecting a neural network model, substituting the model formula into the fractional-order partial differential equation to be solved, a new fractional-order partial differential equation is constructed;

[0008] Converting the fractional-order partial differential equation into an integer-order partial differential equation using fractional-order derivatives that satisfy the chain rule;

[0009] Use symbolic computation tools to extract unknown variables and coefficients from integer-order partial differential equations and generate overdetermined nonlinear algebraic equations;

[0010] Solving the overdetermined nonlinear algebraic equations to obtain parameter weights and biases of the neural network;

[0011] The solution of the fractional-order Riccati equation is combined with a neural network model to optimize and generate analytical solutions to the fractional-order partial differential equation.

[0012] Preferably, the neural network model includes:

[0013] The input layer has two neurons, corresponding to the time variable and the spatial variable respectively;

[0014] The first hidden layer has two neurons, whose activation function is the solution of the fractional Riccati equation;

[0015] The second hidden layer has two neurons, corresponding to the square of the output of the previous layer and itself;

[0016] Output layer, outputs the final solution of the neural network.

[0017] Preferably, the fractional derivative adopts a modified Riemann-Liouville fractional derivative, which is defined as:

[0018] When the order is between 0 and 1, fractional derivatives are defined by a combination of integrals and derivatives, satisfying the chain rule, linearity, and the derivative rule of products.

[0019] Preferably, the properties of the fractional derivative include:

[0020] Derivative the power function, and the fractional derivative value is calculated by the Gamma function;

[0021] Derivative of composite functions, satisfying the chain rule;

[0022] Derivatives of addition and product of functions satisfy linearity and the product rule.

[0023] Preferably, the overdetermined nonlinear algebraic equation system is constructed in the following manner:

[0024] Extract unknown variables and their power coefficients in integer-order partial differential equations;

[0025] Set the coefficients of different powers to 0 to generate an overdetermined system of nonlinear algebraic equations;

[0026] The unknown variables include input time variables and space variables, and implicit variables in the neural network.

[0027] Preferably, the result of solving the overdetermined nonlinear algebraic equations includes the weights and biases of the neural network, specifically:

[0028] Weights and biases from the input layer to the hidden layer;

[0029] The connection weights and biases between neurons within the hidden layer;

[0030] Final weights and biases of the output layer neurons.

[0031] Preferably, the fractional-order Riccati equation has the following characteristics:

[0032] Its solutions can generate generalized hyperbolic function solutions, generalized trigonometric function solutions or rational solutions according to different parameter conditions;

[0033] When the parameter is zero, the resulting solution is a rational solution in the form of a Gamma function that describes the behavior of fractional derivatives.

[0034] Preferably, the solution of the fractional-order Riccati equation is used to optimize a preliminary analytical solution, and a final analytical solution is generated by a combination of weights, biases and the solution of the fractional-order Riccati equation.

[0035] Preferably, the analytical solution includes:

[0036] When the parameter is positive, the analytical solution involves generalized fractional tangent or cotangent functions;

[0037] When the parameter is negative, the analytical solution involves the generalized fractional hyperbolic tangent or hyperbolic cotangent function;

[0038] When the parameter is zero, the analytical solution contains the form of the Gamma function.

[0039] Preferably, the solver is used for multiple fractional-order partial differential equations, including the fractional-order CKdV-mKdV equation, and the generated analytical solutions include generalized fractional-order hyperbolic function solutions, generalized fractional-order trigonometric function solutions and rational solutions.

[0040] The present invention provides a novel fractional sub-equation neural network analytical solver, which has the following beneficial effects:

[0041] 1. The present invention transforms the solution of fractional-order partial differential equations into a parameter solution problem by constructing a neural network model, which significantly reduces the workload of complex mathematical operations in traditional analytical methods. Combining symbolic calculation tools and numerical optimization methods, complex overdetermined nonlinear algebraic equations can be solved quickly, thereby greatly improving the efficiency of solving analytical solutions to fractional-order partial differential equations.

[0042] 2. The present invention uses the special solution of the fractional-order Riccati equation as the activation function, and generates a wider range of analytical solutions through the flexible combination of generalized fractional-order hyperbolic functions, generalized fractional-order trigonometric functions and rational functions. This method can accurately capture the dynamic characteristics of fractional-order systems, expand the scope of application of the solution, and make it applicable to more complex systems.

[0043] 3. The present invention adopts the Jumarie modified Riemann-Liouville fractional derivative that satisfies the chain rule, which ensures the integrity of the non-locality and memory characteristics of the fractional partial differential equation in the analytical solution process. By retaining the mathematical characteristics, the generated analytical solution is not only theoretically consistent, but also meets the accuracy requirements of practical applications.

[0044] 4. The neural network model designed by the present invention combines the dynamic characteristics of fractional-order derivatives and the nonlinear fitting ability of neural networks, and can flexibly adapt to a variety of fractional-order partial differential equations. The multi-layer design of the model not only ensures the accuracy of the calculation, but also can meet the solution requirements of different problems through the adjustment of weights and biases.

[0045] 5. By converting fractional-order partial differential equations into integer-order forms and combining them with generalized function optimization solutions, the present invention can solve a variety of fractional-order partial differential equations including the fractional-order CKdV-mKdV equation. Its versatility and extensibility provide powerful tool support for the study of fractional-order systems.

[0046] 6. The present invention uses symbolic computing tools to extract and process the system of equations, realizing automation from the construction of fractional-order partial differential equations to the generation of analytical solutions. This method reduces the complexity of manual intervention, improves the repeatability and accuracy of the solution process, and provides an efficient solution for engineering and scientific computing.

[0047] 7. The analytical solution generated by the present invention is flexible in form and can be directly used to describe complex dynamic behaviors in materials science, control engineering and biological systems. Its ability to accurately describe nonlinear and nonlocal problems lays the foundation for the application of fractional order theory in practical scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the solution method flow of the solver of the present invention;

[0049] Figure 2 It is a schematic diagram of a neural network model of the present invention;

[0050] Figure 3 A schematic diagram of a selected neural network model of the present invention;

[0051] Figure 4 It is a schematic diagram of the sub-equation neural network method of the present invention. DETAILED DESCRIPTION

[0052] The following will be combined with the drawings of the present invention specification 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.

[0053] Please see attached Figure 1 -Attached Figure 4 The present invention provides a novel fractional sub-equation neural network analytical solver, which can solve the analytical solution of the fractional partial differential equation (fPDE) by combining the neural network and the fractional derivative method. The following is a detailed description of the specific implementation steps and principles of the present invention.

[0054] S1. Select the neural network model and build the formula

[0055] Step S1 is the key foundation of the entire solution process. Its core lies in selecting a suitable neural network model, substituting the model formula into the fractional-order partial differential equation to be solved, and then constructing a new fractional-order partial differential equation, laying the foundation for the analytical transformation and solution of the subsequent steps. Specifically, the selection of the neural network model and the construction of the formula need to fully combine the characteristics of the fractional-order partial differential equation to ensure the adaptability of the model and the operability of the calculation.

[0056] In this embodiment, a suitable neural network model is first selected to adapt to the solution requirements of fractional-order partial differential equations. Specifically, the present invention adopts a 2-2-2-1 type fully connected neural network model, which has the characteristics of simple structure and strong expression ability, and can effectively capture the nonlinear characteristics and dynamic behaviors of fractional-order partial differential equations.

[0057] Generally speaking, the neural network model includes the following parts:

[0058] The input layer contains two neurons, representing the independent variables x (spatial variable) and t (temporal variable).

[0059] The first hidden layer contains two neurons, whose activation function is the solution of the fractional Riccati equation Used to characterize nonlocal properties in fractional-order partial differential equations;

[0060] The second hidden layer contains two neurons, one of which outputs the square of the result of the previous layer, and the other outputs the result of the previous layer itself;

[0061] The output layer contains one neuron, which represents the final analytical solution u.

[0062] As an option, the structure of the network model can be expressed by the following formula:

[0063]

[0064] in, is the solution of the fractional-order Riccati equation, ξ1,ξ2 are linear combinations of input variables, and w i,j represents the network weight and b is the bias.

[0065] Specifically, the activation function The form of is determined by the fractional Riccati equation, which is defined as:

[0066]

[0067] The solutions can be divided into the following cases according to the sign of σ:

[0068] When σ>0, It can be expressed in the form of generalized trigonometric functions, such as tan α (ξ) or cot α (ξ);

[0069] When σ=0, There is a form of Gamma function;

[0070] When σ<0, Can be expressed as a generalized hyperbolic function, such as tanhα (ξ) or coth α (ξ).

[0071] In one possible implementation, a new fractional-order partial differential equation can be constructed by substituting the above network formula into the fractional-order partial differential equation. In general, the fractional-order partial differential equation can be expressed as:

[0072]

[0073] in, 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 α; satisfies 0<α≤1, and f is a function containing u and its fractional derivatives of various orders.

[0074] As a specific implementation, the u and ξ in the network formula are substituted into the above fractional-order partial differential equation one by one, and the hierarchical relationship in the neural network model is used to gradually analyze the dependency structure of the fractional-order derivative terms. For example:

[0075]

[0076] in, is a fractional derivative, which can be further expanded according to the chain rule as follows:

[0077]

[0078] In one possible implementation, in order to ensure the expressiveness of the model, additional activation function forms, such as tanh function and sigmoid function, can be introduced to enhance the adaptability of the model to different types of fractional-order partial differential equations.

[0079] In this implementation, by constructing a neural network model and substituting its formula into the fractional-order partial differential equation to be solved, a new fractional-order partial differential equation is generated, which provides a unified expression for subsequent steps. As an option, the weights and biases of the neural network can be initialized according to the specific problem requirements, or further solved through symbolic calculation tools. This process not only enhances the flexibility of analytical solution of fractional-order partial differential equations, but also provides a mathematical basis for the transformation combined with fractional-order derivatives.

[0080] S2. Use fractional derivatives to convert to integer-order partial differential equations

[0081] The goal of step S2 is to convert the fractional-order partial differential equation constructed by the neural network into an integer-order partial differential equation using fractional-order derivatives that satisfy the chain rule. Fractional-order derivatives have non-local characteristics and can more accurately describe the dynamic behavior of the system, while their chain rule and linearity make it possible to analytically solve fractional-order partial differential equations. Through this step, complex fractional-order problems are converted into more easily manipulated integer-order forms, providing a computational basis for subsequent steps.

[0082] In this implementation, the fractional-order partial differential equation is analytically transformed using the Jumarie-corrected Riemann-Liouville fractional-order derivative.

[0083] In general, the Jumarie modified Riemann-Liouville fractional derivative is defined as follows:

[0084] For fractional orders 0<α<1, its form is:

[0085]

[0086] Where Γ(z0) is the Gamma function and f(0) is the value of the function at the starting point.

[0087] When α is a positive integer or non-integer, the definition of fractional derivative is unified as follows:

[0088]

[0089] As an option, fractional derivatives satisfy the following properties:

[0090] Linearity:

[0091]

[0092] Chain Rule:

[0093] For the composite function f[g(x)], we have:

[0094]

[0095] Derivative of power function:

[0096]

[0097] In one possible implementation, combined with the hierarchical expression of the neural network, the chain expansion of the fractional derivative on the activation function is:

[0098]

[0099] in, is the form of the activation function, and ξ is a linear combination of the input variables.

[0100] Specifically, using the above properties, the fractional-order partial differential equation constructed by the neural network can be transformed into an integer-order partial differential equation. For example, for the fractional-order partial differential equation:

[0101]

[0102] Using the Network Formula Its fractional derivative term can be expanded as:

[0103]

[0104] In one possible implementation, the definition of the fractional-order Riccati equation is combined with Further using the chain rule, and Respectively expressed as:

[0105]

[0106] Thus, the new fractional-order partial differential equation is transformed into an integer-order form. In general, the integer-order partial differential equation can be simplified as:

[0107] u t +ηuu x +βu 2 u x +γu xxx =0,

[0108] Among them, u t ,u x ,u xxx is an integer-order derivative term, and η, β, γ are combinations of fractional-order derivative terms and weight parameters.

[0109] In this implementation, by combining the definition and properties of Jumarie's modified Riemann-Liouville fractional derivatives, the complex non-local problems in the fractional partial differential equations are gradually transformed into local problems in integer order form. As an option, the chain rule in the transformation process ensures the differentiability between the neural network levels, and the linearity of the fractional derivatives further simplifies the construction of the expression. In practical applications, this process can be automated in combination with symbolic computing tools, thereby providing an efficient mathematical basis for the generation of subsequent analytical solutions.

[0110] S3. Extract unknown variables and construct equations

[0111] The purpose of step S3 is to use symbolic computing tools to extract unknown variables and their coefficients in integer-order partial differential equations and generate overdetermined nonlinear algebraic equations. Through this step, the complex partial differential equation form can be converted into the form of algebraic equations, providing a basis for the specific solution of parameters in subsequent steps. Symbolic computing tools can efficiently process complex variable relationships and expressions, and by extracting and reducing variable coefficients, the construction of overdetermined nonlinear equations is more systematic and accurate.

[0112] In this embodiment, a symbolic computing tool (such as Maple) is used to extract unknown variables and their coefficients in the integer-order partial differential equation. The use of symbolic computing tools is helpful in processing complex partial differential equations, especially after the fractional-order partial differential equation is converted into an integer-order partial differential equation in step S2, its structure may contain multiple implicit variables and nonlinear terms.

[0113] In general, integer-order partial differential equations are obtained by expanding fractional-order derivatives, and their form usually contains multiple parameters and unknown variables. For example, in the transformed equation:

[0114] u t +ηuu x +βu 2 u x +γu xxx =0,

[0115] Among them, u t ,u x ,u xxx Represent the integer derivatives of time, space and higher-order space respectively, η, β, γ are the unknown parameters in the fractional-order partial differential equation. These parameters are closely related to the weights and biases of the neural network model, such as w t,1 ,w x,1 ,w 3,u ,w 4,u ,b, etc.

[0116] Specifically, through symbolic calculation tools, the unknown variables and parameters in the partial differential equation can be expanded to various powers and the coefficients of each term can be extracted. For the coefficients of different powers, their values ​​are set to zero, thereby generating an overdetermined nonlinear algebraic system of equations.

[0117] As an alternative, the neural network formula After substituting into the integer-order partial differential equation, it can be expanded as:

[0118]

[0119] In one possible implementation, the following unknown variables and their coefficients can be extracted:

[0120] Input variables x and t;

[0121] Intermediate variables of activation function and

[0122] The weight parameter w in the neural network t,1 ,w x,1 ,w t,2 ,w x,2 ,w 3,u ,w 4,u ,w 1,3 ,w 2,3 ,w 1,4 ,w 2,4 ;

[0123] Bias b.

[0124] By sorting out the different powers of these variables, the overdetermined nonlinear algebraic equations are constructed in the form of:

[0125]

[0126] The maple tool can be used to efficiently simplify and reduce the above algebraic expressions and extract the coefficients of each term.

[0127] As a specific implementation, the symbolic computing tool can further organize the generated equations into standardized expressions. For example:

[0128]

[0129] In this implementation, through the use of symbolic computing tools, the complex partial differential equation form is gradually converted into an overdetermined nonlinear algebraic equation system. In general, this process can effectively separate unknown variables and their associated coefficients, and provide a mathematical expression that can be directly solved for subsequent steps. As an option, by increasing the degree of freedom of the variables (such as introducing high-order derivative terms), the adaptability and accuracy of the equation system can be further enhanced. This method is not only universal in fractional partial differential equations, but can also be extended to the analytical solution of other complex differential equations.

[0130] S4. Solving overdetermined nonlinear algebraic equations

[0131] The goal of step S4 is to solve the overdetermined nonlinear algebraic equations to obtain the parameter weights and biases of the neural network. This step is the core of realizing the training of the neural network model. By converting the complex fractional-order partial differential equation problem into a mathematical optimization problem, it can provide accurate parameter support for the neural network and further generate a preliminary analytical solution.

[0132] In this embodiment, an overdetermined nonlinear algebraic equation system is generated from step S3, which includes the weights and biases of each layer of the neural network and the undetermined parameters of the fractional-order partial differential equation. By solving these equations, the specific parameter values ​​of the neural network model can be determined and substituted into the network formula to generate a preliminary analytical solution.

[0133] Typically, overdetermined nonlinear algebraic equations consist of relationships between variables and coefficients extracted by symbolic computation tools, for example:

[0134]

[0135] Among them, w t,2 ,w x,2 ,γ,η,b, etc. are network parameters or unknown constants in partial differential equations.

[0136] Specifically, the overdetermined nonlinear algebraic equations can be expressed as a set of implicit equations:

[0137] F1(wt,1,w x,1 ,η,β,γ)=0,

[0138] F2(w t,2 ,w x,2 ,η,β,γ)=0,

[0139] F3(w 1,3 ,w 2,3 ,w 3,u ,η,β)=0,

[0140] ···

[0141] The structure of these systems of equations reflects the complex dependencies between the network weights.

[0142] In one possible implementation, the above-mentioned overdetermined nonlinear algebraic equations are solved using a symbolic computing tool (such as Maple or Mathematica). In general, the symbolic computing tool can be processed by the following method:

[0143] The relationship between parameters is solved by symbolic computing software;

[0144] Bringing these relationship solutions back to the neural network model, we can get a preliminary solution, which is the solution to the partial differential equation.

[0145] In some embodiments, the overdetermined nonlinear algebraic equation system may include up to 36 equations, corresponding to different powers of variables and different combinations of parameters. For example:

[0146] The coefficient of the first-order term corresponding to the variable x;

[0147] The coefficient of the quadratic term corresponding to the variable t;

[0148] Constant term F b The optimal condition for the corresponding bias.

[0149] By gradually solving the set of equations, we can obtain the fully connected parameters of the neural network model, including the weights from the input layer to the hidden layer, the weights between the hidden layers, and the bias of the output layer. Specifically, the network parameters after the solution are in the form of:

[0150] α=α, β=β, b=b, η=η,

[0151] w x,1 =w x,1 ,w x,2 =w x,2 ,w 1,3 =w 1,3 ,w 1,4 =w 1,4 ,

[0152] w 2,3 =w 2,3 ,w 2,4 =w 2,4 ,w 3,u =w 3,u ,w 4,u =w 4,u ,

[0153]

[0154] In one possible implementation, the solved weights and biases are substituted into the neural network formula:

[0155]

[0156] The specific expressions of ξ3 and ξ4 are determined by the network input and activation function. Further expansion of the formula can generate a preliminary analytical solution.

[0157] In this embodiment, the optimization of the neural network model parameters is completed by solving the overdetermined nonlinear algebraic equations, and a preliminary analytical solution is generated. In general, the solution process of this step can be combined with symbolic calculation and numerical optimization methods to ensure the stability and accuracy of parameter solution. As an option, the degree of freedom of the system of equations can be adjusted according to the complexity of the specific fractional-order partial differential equation to meet the needs of solving a wider range of problems. This method is not only applicable to the fractional-order problems in the present invention, but can also be extended to the generation of analytical solutions to other nonlinear differential equations.

[0158] S5. Combine the fractional-order Riccat i equation to obtain an analytical solution

[0159] The core of step S5 is to combine the solution of the fractional-order Riccati equation and the neural network parameters obtained in step S4 to obtain the final analytical solution of the fractional-order partial differential equation. By introducing the solution of the fractional-order Riccati equation, this step can effectively enhance the expression ability of the solution while ensuring its mathematical completeness.

[0160] In this implementation, the form of the fractional-order Riccati equation is defined as:

[0161]

[0162] in, represents the fractional derivative, and σ is a constant. According to the sign characteristics of σ, the solutions of the fractional Riccati equation can be divided into the following cases:

[0163] When σ>0, the solution is in the form of generalized trigonometric functions, such as tan α (ξ) or cot α (ξ);

[0164] When σ<0, the solution is in the form of a generalized hyperbolic function, such as tanh α (ξ) or coth α (ξ);

[0165] When σ = 0, the solution is in the form of a Gamma function, which represents a special fractional-order function.

[0166] In general, the specific expression of generalized fractional trigonometric and hyperbolic functions can be expressed by the Mittag-Leffler function E α (z). For example:

[0167]

[0168] As an option, the preliminary analytical solution can be further optimized by substituting the solution of the fractional-order Riccati equation into the neural network formula obtained in step S4. Specifically, the output formula of the neural network is:

[0169]

[0170] in:

[0171]

[0172] is the solution of the fractional-order Riccati equation.

[0173] In one possible implementation, the neural network formula is combined with the solution of the fractional-order Riccati equation according to the sign of σ to generate the final analytical solution. For example, when σ>0, the analytical solution is in the form of:

[0174]

[0175] Where ψ = tw t,1 +xw x,1 ,

[0176] Specifically, when σ = 0, the analytical solution is in the form of:

[0177]

[0178] In a specific implementation, in order to further improve the accuracy of the analytical solution, additional adjustment factors can be introduced according to the problem requirements. For example, adjustment weights are introduced into the parameters of the fractional-order Mittag-Leffler function so that the characteristics of the solution can better fit the dynamic characteristics of the actual problem.

[0179] In this embodiment, by combining the solution of the fractional-order Riccati equation, the preliminary analytical solution is optimized, and the final analytical solution of the adapted fractional-order partial differential equation is generated. In general, this process can effectively combine the dynamic characteristics of the fractional-order derivatives with the nonlinear expression ability of the neural network model. As an option, the form of the analytical solution can be adjusted according to the needs of specific problems, such as introducing more generalized functions or piecewise defined solution forms to adapt to different fractional-order partial differential equation problems. This method is not only applicable to the specific problems in the present invention, but also has certain generalizability and can be applied to the analytical solution of other complex dynamic systems.

[0180] In general, the present invention constructs a neural network model, substitutes its formula into a fractional-order partial differential equation, uses the Jumarie modified Riemann-Liouville fractional-order derivative that satisfies the chain rule to convert the problem into an integer-order form, uses symbolic calculation tools to extract unknown variables to generate an overdetermined nonlinear algebraic equation system, and obtains the weights and biases of the neural network by solving the equation system. Finally, by optimizing the preliminary analytical solution in combination with the solution of the fractional-order Riccati equation, a final analytical solution in the form of a generalized hyperbolic function, a generalized fractional-order trigonometric function, and a Mittag-Leffler function is generated. The present invention has high efficiency and versatility, and provides a new method for solving fractional-order partial differential equations.

[0181] In order to better understand the present invention, the above method is described in detail below in conjunction with specific embodiments.

[0182] Example:

[0183] The embodiment of the present invention provides a novel fractional-order sub-equation neural network analytical solver, comprising the following steps:

[0184] Step 1: According to the needs, select a suitable neural network model and write its corresponding formula, substitute the model formula into the fractional-order partial differential equation to be calculated, in this way, a new fractional-order partial differential equation is obtained.

[0185] The neural network model selected by the present invention is a 2-2-2-1 model, that is, the input layer has two neurons, x and t, and the activation functions corresponding to the two neurons in the first hidden layer are both the fractional-order Riccati equation The activation functions corresponding to the two neurons in the third layer are themselves and the square term respectively. The last layer is the output layer, corresponding to u. The specific formula is as follows:

[0186]

[0187] Step 2: Convert the new fractional-order PDE to an integer-order PDE by using the Riemann-Liouville fractional-order derivative modified by Jumarie, so as to satisfy the chain rule similar to that of integer-order derivatives.

[0188] The modified Riemann-Liouville fractional derivative is defined as:

[0189]

[0190] The properties are:

[0191]

[0192] Using the above definitions and properties, the integer-order partial differential equation is:

[0193]

[0194]

[0195] Since the form of integer-order partial differential equations is relatively complicated, it is abbreviated here.

[0196] Step 3: Find the unknown variables and use the symbolic computing software Maple to merge the coefficients of the same power of these unknown variables, and set the coefficients of the different powers of the unknown variables to zero. This will give an overdetermined nonlinear algebraic equation system about the coefficients of the integer-order partial differential equation. The unknown variables of this equation are: x, t,

[0197] The 36 sets of overdetermined nonlinear algebraic equations solved by Maple are:

[0198]

[0199]

[0200]

[0201]

[0202]

[0203] Step 4: Solve the overdetermined nonlinear algebraic equations, find the fully connected neural network in the obtained solution, and substitute it into the selected tensor formula to obtain a preliminary analytical solution.

[0204] The fully connected neural network is:

[0205] α=α,β=β,b=b,η=η,w x,1 =w x,1 ,w x,2 =w x,2 ,w 1,3 =w 1,3 ,w 1,4 =w 1,4 ,

[0206] w 2,3 =w 2,3 ,w 2,4 =w 2,4 ,w 3,u =w 3,u ,w 4,u =w 4,u ,w t,1 =w t,1 ,

[0207] Step 5: According to the fractional order Riccati equation: The solutions under different symbols are combined with the preliminary analytical solution obtained in step 4 to determine the final analytical solution u.

[0208] The final analytical solution is as follows:

[0209] Solution 1: σ>0.

[0210]

[0211] Solution 2: σ>0.

[0212]

[0213] Solution 3: σ=0.

[0214]

[0215] Solution 4: σ<0.

[0216]

[0217] Solution 5: σ<0.

[0218]

[0219] Where ψ = tw t,1 +xw x,1 ,

[0220] 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 neural network analytical solver for fractional sub-equations, characterized by: The following steps are involved: By selecting a neural network model, substituting the model formula into the fractional-order partial differential equation to be solved, a new fractional-order partial differential equation is constructed; Converting the fractional-order partial differential equation into an integer-order partial differential equation using fractional-order derivatives that satisfy the chain rule; Use symbolic computation tools to extract unknown variables and coefficients from integer-order partial differential equations and generate overdetermined nonlinear algebraic equations; Solving the overdetermined nonlinear algebraic equations to obtain parameter weights and biases of the neural network; The solution of the fractional-order Riccati equation is combined with a neural network model to optimize and generate analytical solutions to the fractional-order partial differential equation.

2. A novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The neural network model includes: The input layer has two neurons, corresponding to the time variable and the spatial variable respectively; The first hidden layer has two neurons, whose activation function is the solution of the fractional Riccati equation; The second hidden layer has two neurons, corresponding to the square of the output of the previous layer and itself; Output layer, outputs the final solution of the neural network.

3. A novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The fractional derivative adopts the modified Riemann-Liouville fractional derivative, which is defined as: When the order is between 0 and 1, fractional derivatives are defined by a combination of integrals and derivatives, satisfying the chain rule, linearity, and the derivative rule of products.

4. A novel fractional sub-equation neural network analytical solver according to claim 3, characterized in that: The properties of the fractional derivatives include: Derivative the power function, and the fractional derivative value is calculated by the Gamma function; Derivative of composite functions, satisfying the chain rule; Derivatives of addition and product of functions satisfy linearity and the product rule.

5. The novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The overdetermined nonlinear algebraic equations are constructed in the following way: Extract unknown variables and their power coefficients in integer-order partial differential equations; Set the coefficients of different powers to 0 to generate an overdetermined system of nonlinear algebraic equations; The unknown variables include input time variables and space variables, and implicit variables in the neural network.

6. A novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The results of solving the overdetermined nonlinear algebraic equations include the weights and biases of the neural network, specifically: Weights and biases from the input layer to the hidden layer; The connection weights and biases between neurons within the hidden layer; Final weights and biases of the output layer neurons.

7. The novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The fractional-order Riccati equation has the following characteristics: Its solution can generate generalized fractional-order hyperbolic function solution, generalized fractional-order trigonometric function solution or rational solution according to different parameter conditions; When the parameter is zero, the resulting solution is a rational solution in the form of a Gamma function that describes the behavior of fractional derivatives.

8. The novel fractional sub-equation neural network analytical solver according to claim 7, characterized in that: The solution of the fractional-order Riccati equation is used to optimize the preliminary analytical solution, and the final analytical solution is generated by a combination of weights, biases and the solution of the fractional-order Riccati equation.

9. A novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The analytical solution takes the form of: When the parameter is positive, the analytical solution involves generalized fractional tangent or cotangent functions; When the parameter is negative, the analytical solution involves the generalized fractional hyperbolic tangent or hyperbolic cotangent function; When the parameter is zero, the analytical solution contains the form of the Gamma function.

10. The novel fractional sub-equation neural network analytical solver according to claim 1, characterized in that: The solver is used for multiple fractional-order partial differential equations, including the fractional-order CKdV-mKdV equation, and the generated analytical solutions include generalized fractional-order hyperbolic function solutions, generalized fractional-order trigonometric function solutions and rational solutions.

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