Neutron diffusion simulation method based on improved PINNs

Abstract

A neutron diffusion simulation method based on improved PINNs (Physical Information Neural Networks) is proposed. This method utilizes an improved PINN to obtain the specific neutron flux distribution of the geometric model to be solved and acquire the corresponding eigenvalues, thereby achieving neutron diffusion simulation calculations. This invention can calculate neutron diffusion equations for complex problems involving vacuum boundaries, contact boundaries, and eigenvalues ​​used in practical engineering problems. It provides a high-precision and high-efficiency computational method for calculating and predicting neutron behavior inside reactors in practical engineering problems.

Description

Technical Field

[0001] This invention relates to a technology in the field of reactor simulation, specifically a neutron diffusion simulation method based on an improved physical information neural network (PINNs). Background Technology

[0002] While existing neural networks and Physical Information Neural Networks (PINNs) have made significant progress in solving differential eigenvalue equations, they cannot yet be directly applied to solving neutron diffusion equations in nuclear engineering. Firstly, engineering models are geometrically complex, and certain boundary conditions can worsen the PINN calculation of eigenfunctions. Furthermore, large-scale engineering numerical modeling often involves contact boundary conditions with various materials and media, requiring an efficient and accurate method for calculating these boundaries using PINNs. Secondly, the neutron diffusion equation, as a typical eigenvalue equation, cannot be directly solved using PINNs. Obtaining the eigenfunctions and their corresponding eigenvalues ​​still requires high-precision and efficient methods. Additionally, when using PINNs to calculate eigenvalue equations for engineering applications, the specific constraints of eigenvalues ​​and boundary conditions often prevent them from moving away from trivial or zero solutions during training. Summary of the Invention

[0003] To address the aforementioned shortcomings of existing technologies, this invention proposes a neutron diffusion simulation method based on improved PINNs. Utilizing the improved PINNs, it is possible to calculate neutron diffusion equations for complex problems involving vacuum boundaries, contact boundaries, and eigenvalues ​​in practical engineering applications. This provides a high-precision and high-efficiency computational method for calculating and predicting neutron behavior inside reactors in practical engineering problems.

[0004] This invention is achieved through the following technical solution:

[0005] This invention relates to a neutron diffusion simulation method based on improved PINNs, comprising:

[0006] Step 1: Construct an improved physical information neural network;

[0007] Step 2: After determining the geometric model, boundary conditions, equation parameters and number of computational domains of the neutron diffusion eigenvalue equation to be solved, select training points in the computational domain, input the training point data into the fully connected layer to obtain the trial function of the neutron diffusion equation solution and create the boundary condition loss.

[0008] Step 3: Based on the training points, perform hard-constrained fixed-point transformation or soft-constrained fixed-point transformation on the trial function obtained in Step 2 to obtain the fixed-point loss, and construct the PDE loss through automatic differentiation.

[0009] Step 4: Select training points on the contact boundary and obtain the contact boundary loss by splicing the computational domains;

[0010] Step 5: Set up training points on the vacuum boundary, and calculate the vacuum extrapolation boundary loss by extrapolating the boundary from the vacuum boundary, i.e., setting up training points for the extrapolation boundary according to the extrapolation distance.

[0011] Step 6: Summarize the PDE loss, boundary condition loss, fixed point loss, contact boundary loss, and vacuum extrapolation boundary loss to obtain the total loss function, and train the improved physical information neural network using the Adam or L-BFGS optimizer.

[0012] The summary mentioned above refers to: the total loss function ,in: The corresponding weight for each individual loss term, The boundary condition loss obtained in step 2, and These are the fixed-point loss and PDE loss obtained in step 3, respectively. , These are the contact boundary losses obtained in step 4. This is the vacuum extrapolation boundary loss obtained in step 5.

[0013] Step 7: In the online phase, the neutron flux distribution of the geometric model to be solved is output through the fully connected layer of the trained improved physical information neural network, and the corresponding feature values ​​are obtained to realize the neutron diffusion simulation calculation.

[0014] The improved physical information neural network comprises: a fully connected layer, a fixed-point transformation layer, an automatic differential layer, a computational domain splicing layer, and a vacuum extrapolation boundary layer. Specifically: the fully connected layer reads training points based on trainable parameters such as weights, biases, and additional trainable parameters representing predicted eigenvalues, solves the neutron diffusion equation to obtain a trial function, and calculates the boundary condition loss; the fixed-point transformation layer performs a fixed-point transformation on the trial function to obtain a trial function constrained by the fixed-point transformation, and creates a fixed-point loss; the automatic differential layer automatically differentiates the trial function updated by the fixed-point transformation layer based on the time and spatial coordinates of the training points, creating a PDE loss; the computational domain splicing layer divides the trial function constrained by the fixed-point transformation according to the computational domains of different materials used in the physical model, and creates a contact boundary loss by reading additionally set contact boundary training points; the vacuum extrapolation boundary layer performs boundary extrapolation based on the training points of the vacuum boundary, creating a vacuum extrapolation boundary loss.

[0015] Technical effect

[0016] This invention employs a fixed-point transformation, computational domain splicing, and virtual vacuum boundary training point method to address practical neutron computation engineering problems. This enables the improved PINNs to perform neutron diffusion eigenvalue equation calculations with contact and vacuum boundaries based on the input geometric model. For contact boundary problems, a computational domain splicing method is used to calculate eigenvalue equations for different materials, and the calculation of contact boundaries is completed based on boundary conditions commonly used in engineering. For solving eigenfunctions and eigenvalues, a fixed-point transformation method is constructed. By setting fixed points in the computational domain, the correct eigenvalue values ​​can be obtained while simultaneously acquiring the eigenfunction distribution. To address the deterioration of the PINNs computation process by third-type boundaries such as vacuum boundaries, a virtual vacuum boundary training point method is used to convert third-type boundaries into first-type boundaries, optimizing the training effect of PINNs. Attached Figure Description

[0017] Figure 1 This is a flowchart of the present invention;

[0018] Figure 2 This is a flowchart of an implementation example;

[0019] Figure 3 This is a schematic diagram of a numerical experiment on the one-dimensional neutron diffusion equation using the improved PINNs in the embodiment.

[0020] Top left in the figure: neutron distribution results; top right: absolute error between neutron distribution and true value; bottom left: neural network training loss descent curve; bottom right: eigenvalue calculation convergence curve;

[0021] Figure 4 This is a schematic diagram of a numerical experiment on the two-dimensional neutron diffusion equation using the improved PINNs in the embodiment.

[0022] In the figure: Top: Numerical solution results for neutron distribution; Middle left: Neutron distribution results and absolute error for hard-constrained PINNs; Middle right: Neutron distribution results and absolute error for soft-constrained PINNs; Bottom left: Neural network training loss descent curve; Bottom right: Eigenvalue calculation convergence curve;

[0023] Figure 5 This is a schematic diagram of the numerical experiment on the neutron distribution in a loosely coupled reactor using the improved PINNs in the example.

[0024] In the figure: Top left: Geometry of loosely coupled reactor; Top right: Numerical solution results of neutron distribution; Middle left: Neutron distribution results and absolute error of hard-constrained PINNs; Middle right: Neutron distribution results and absolute error of soft-constrained PINNs; Bottom left: Neural network training loss descent curve; Bottom right: Eigenvalue calculation convergence curve. Detailed Implementation

[0025] This embodiment relates to a neutron diffusion simulation system based on improved PINNs, including: a fully connected layer module, an automatic differentiation module, a loss function module, a fixed-point transformation module, a computational domain splicing module, and a vacuum extrapolation boundary module. Specifically: the fully connected layer module reads training points based on trainable parameters such as weights, biases, and additional trainable parameters representing eigenvalue predictions, solves the neutron diffusion equation to obtain a trial function, and calculates the boundary condition loss; the fixed-point transformation module performs a fixed-point transformation on the trial function to obtain a trial function with fixed-point constraints and creates a fixed-point loss; the automatic differentiation module calculates the loss based on the time-varying training points. The system uses spatial coordinates to automatically differentiate the trial function updated by the fixed-point transformation module, creating the PDE loss. The computational domain stitching module divides the trial function after the fixed-point transformation constraint according to the computational domain of different materials used in the physical model, and creates the contact boundary loss by reading the additionally set contact boundary training points. The vacuum extrapolation boundary module performs boundary extrapolation based on the training points of the vacuum boundary, creating the vacuum extrapolation boundary loss. The loss function module constructs the total loss function based on the PDE loss, boundary condition loss, fixed-point loss, contact boundary loss, and vacuum extrapolation boundary loss, which is used to train the improved PINNs.

[0026] like Figure 1 As shown, this embodiment relates to a neutron diffusion simulation method based on the above system, including:

[0027] Step 1: Construct the above-mentioned improved physical information neural network (PINNs).

[0028] Step 2: After determining the geometric model, boundary conditions, equation parameters, and number of computational domains for the neutron diffusion eigenvalue equations to be solved, select training points within the computational domains. and The training data is input into the fully connected layer module to obtain the trial function of the neutron diffusion equation solution and to create the boundary condition loss, specifically including:

[0029] 2.1 Based on the fact that the calculated geometric model contains two computational domains, the output trial function of the neutron diffusion equation in computational domain 1 is... With the output trial function of computation domain 2 ,in: For the domain coordinates, For training points of PDE loss, These are the training points for the boundary loss.

[0030] 2.2 Based on the boundary conditions of the geometric model and the trial function of the output, create a boundary condition loss function. ,in: Differential operators used to represent boundary conditions This represents computational domain 1 or 2. For the boundary condition loss of computational domain 1, For the boundary condition loss of computation domain 2, .

[0031] Step 3: Using the fixed-point transformation module, the coordinates of the fixed points selected in computation domain 1 are transformed. With numerical values After performing hard-constrained fixed-point transformation or soft-constrained fixed-point transformation on the trial function obtained in step 2, the PDE loss is constructed through the automatic differentiation module.

[0032] The aforementioned hard constraint fixed-point transformation refers to directly transforming the output of the constraint PINNs through a function transformation, specifically: for Perform the transformation: ,in: Using fixed point coordinates, the output of PINNs is... The value at the fixed point is Subsequently, all loss functions were calculated using... Replace the original .

[0033] The soft-constraint fixed-point transformation refers to adding an additional fixed-point loss term to the loss function module in step 1, specifically: This makes in Point output It can continuously move towards a constant as the loss function decreases. To move closer.

[0034] The fixed-point transformation is equivalent to adding additional physical constraints to PINNs. These fixed-point physical constraints make the training process of PINNs more controllable, enabling not only... Avoiding trivial solutions can also alleviate the deterioration of PINNs training caused by various boundary conditions.

[0035] The PDE loss is specifically as follows: ,in: This represents the operator used in the neutron diffusion equation of the computational model. Additional trainable parameters for setting representative feature value predictions for fully connected layer modules. This represents computation domain 1 or 2. For the PDE loss in computation domain 1, For the PDE loss in computation domain 2, .

[0036] Step 4: Select training points on the contact boundary. The input is then processed into the computational domain stitching module to obtain the contact boundary loss function, specifically: ,in: , These are the two outputs of the fully connected layer, used to fit the computational domain. Distribution of physical field characteristic functions within and computational domain Distribution of physical field characteristic functions within , The trainable parameters representing the feature values, These are the trainable parameters for the fully connected layer modules. and These are the diffusion coefficients of the materials on both sides, Training points used to contact the boundary.

[0037] Step 5: Set training points on the vacuum boundary, input the vacuum extrapolation boundary module, set the training points for the extrapolation boundary according to the extrapolation distance, and calculate the vacuum extrapolation boundary loss. Specifically, when the vacuum boundary is located within computational domain 1, the vacuum extrapolation boundary loss is... ,in: The training points are extrapolated from the vacuum boundary, and their calculation formula is as follows: ,and , This represents the diffusion coefficient of the material at the extrapolated boundary.

[0038] Step 6: The loss function module reads the boundary condition loss from Step 2, the fixed-point loss and PDE loss from Step 3, the contact boundary loss from Step 4, and the vacuum extrapolation boundary loss from Step 5 to obtain the total loss function. The total loss function is then input into the Adam or L-BFGS optimizer, and training is completed after the set number of training steps. The neutron flux distribution (feature function) of the geometric model is set through the output of the trained fully connected layer, along with additional trainable parameters in the fully connected layer module. These are eigenvalues, specifically including:

[0039] 6.1 Set the training steps, optimizer, and learning rate as shown in Table 1.

[0040] Table 1

[0041]

[0042] 6.2 Before reaching the required number of training steps, optimize the trainable parameters of the fully connected layer by calculating the total loss function, specifically as follows: ,in: The weights for each individual loss term are determined as follows: Normally, all weights can be set to 1, but in some special cases, certain weights are increased according to the training process. For example, in scenario 2, due to the smaller computational area, the weights are appropriately increased. The value is 10. In scenario 5, due to the large computational region and relatively small fixed-point constraint strength, the improvement is... It is 10.

[0043] Step 7: In the online phase, the trial function output by the fully connected layer represents the neutron flux distribution (characteristic function) results in each computational domain, with additional trainable parameters. This represents the final eigenvalue result. Finally, the neutron flux distribution (eigenfunction) results in each computational domain are spliced ​​together at the contact boundary to obtain the calculation results of the neutron diffusion equation as shown in Tables 2-4.

[0044] Table 2 Calculation results of neutron diffusion equation in scenario 2

[0045]

[0046] Table 3 Calculation results of neutron diffusion equations in scenario 4

[0047]

[0048] Table 4 Calculation results of neutron diffusion equation in scenario 5

[0049]

[0050] Unlike existing PINNs, which can only solve relatively simple PDEs but cannot solve eigenvalues, calculate contact boundaries with high precision, or avoid the degradation of the training process by third-type boundaries (including vacuum boundaries), PINNs employ a fixed-point transformation in step 3 to solve eigenvalues, a computational domain concatenation in step 4 to solve contact boundaries with high precision, and a virtual extrapolation training point in step 5 to solve vacuum boundaries. This gives PINNs the ability to fully compute neutron diffusion equations, especially those considering real-world engineering problems.

[0051] In summary, this invention can calculate the characteristic function (neutron flux density) and eigenvalue (effective multiplication factor) with high precision. Traditional PINNs cannot simultaneously obtain the characteristic function and eigenvalue. The fixed-point transformation method adds extra physical information to the PINNs by introducing additional fixed points into the computational domain. This transforms the solution of the neutron diffusion equation from a forward to a backward approach, allowing for the calculation of the equation's eigenvalues ​​while obtaining the correct neutron flux distribution, and enabling high-precision calculation of the contact boundary. Traditional PINNs cannot solve the case where two computational domains are in contact. The computational domain splicing method uses the same hidden layer parameters to control multiple outputs of the PINNs to represent the neutron flux distribution in different material computational domains. After setting an additional loss function at the contact boundary, splicing the computational domains allows for high-precision calculation of the correct contact boundary result. Furthermore, this invention can accurately fit vacuum boundaries. For traditional PINNs, third-type boundary forms such as vacuum boundaries may degrade the training process and lead to incorrect results. The additional constraints of extrapolated virtual boundaries can optimize the computational results of PINNs.

[0052] The above-described specific implementations can be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of the present invention. The scope of protection of the present invention is defined by the claims and is not limited to the above-described specific implementations. All implementation schemes within the scope of the claims are bound by the present invention.

Claims

1. A method for neutron diffusion simulation based on improved PINNs, characterized in that, The method comprises the following steps: Step 1, building the improved physical information neural network; Step 2, after determining the geometric model, boundary conditions, equation parameters and the number of calculation domains of the neutron diffusion eigenvalue equation to be solved, selecting training points in the calculation domain, inputting the training point data into the full connection layer to obtain a trial function of the neutron diffusion equation solving result and creating a boundary condition loss; Step 3, after the trial function obtained in step 2 is subjected to a hard constraint fixed point transformation or a soft constraint fixed point transformation according to the training points, a fixed point loss is created, and a PDE loss is constructed through automatic differentiation; Step 4, selecting training points on the contact boundary, and obtaining a contact boundary loss through calculation domain splicing; Step 5, setting training points on the vacuum boundary, and calculating a vacuum extrapolation boundary loss through vacuum extrapolation boundary; Step 6, obtaining a total loss function by summing up the boundary condition loss, the PDE loss, the fixed point loss, the contact boundary loss and the vacuum extrapolation boundary loss, and training the improved physical information neural network through an Adam or L-BFGS optimizer; Step 7, in the online phase, outputting the specific results of the neutron flux distribution of the geometric model to be solved through the full connection layer of the trained improved physical information neural network and obtaining the corresponding eigenvalue, so as to realize the neutron diffusion simulation calculation; The improved physical information neural network comprises a full connection layer, a fixed point transformation layer, an automatic differentiation layer, a calculation domain splicing layer and a vacuum extrapolation boundary layer, wherein: the full connection layer reads the training points according to the weights, the bias and the additional trainable parameters representing the eigenvalue prediction value, solves the neutron diffusion equation to obtain a trial function, and creates a boundary condition loss; the fixed point transformation layer performs fixed point transformation on the trial function to obtain a trial function subjected to fixed point transformation constraint and creates a fixed point loss; the automatic differentiation layer performs automatic differentiation on the trial function updated by the fixed point transformation layer according to the training points of time, space coordinates, creates a PDE loss; the calculation domain splicing layer divides the trial function subjected to fixed point transformation constraint according to the calculation domains of different materials of the physical model, creates a contact boundary loss by reading the additional contact boundary training points; and the vacuum extrapolation boundary layer performs boundary extrapolation according to the training points of the vacuum boundary to create a vacuum extrapolation boundary loss; The method is used for neutron diffusion simulation in the field of nuclear engineering.

2. The improved PINNs-based simulation method of neutron diffusion according to claim 1, characterized in that, The step 2 specifically comprises: 2.1 If the number of computational domains contained in the geometric model to be computed is 2, the output trial function of the neutron diffusion equation of computational domain 1 is calculated and the output trial function of computational domain 2 where: are trainable parameters of the fully connected layer, are training points for the PDE loss; 2.2 Create boundary condition loss according to boundary condition of geometry model and output trial function wherein: is the domain coordinate, is the training point of boundary loss, represents the differential operator used by the boundary condition, represents the calculation domain 1 or 2; For the boundary condition loss of the computational domain 1, For the boundary condition loss of the computational domain 2, .

3. The improved PINNs-based simulation method of neutron diffusion according to claim 2, characterized in that, The hard constraint fixed point transformation refers to directly transforming the output result of the PINNs through a function transformation, specifically, transforming wherein is the fixed point coordinate, so that the output result of the PINNs is the value at the fixed point , and then each item of the loss function is calculated using instead of the original .​ 4. The improved PINNs-based simulation method of neutron diffusion according to claim 2, characterized in that, The soft-constraint fixed-point transformation refers to adding an additional fixed-point loss term to the loss function module in step 1, specifically: This makes in Point output It can continuously move towards a constant as the loss function decreases. To move closer.

5. The improved PINNs-based simulation method of neutron diffusion according to claim 2, characterized in that, The PDE loss, in particular: wherein: represents a neutron diffusion equation operator used by the computational model, an additional trainable parameter set for the fully connected layer module representing the eigenvalue prediction, represents the computational domain 1 or 2; For the PDE loss for domain 1, For the PDE loss for domain 2, .

6. The improved PINNs-based simulation method of neutron diffusion according to claim 2, characterized in that, The contact boundary loss mentioned above is specifically as follows: ,in: , These are the two outputs of the fully connected layer, used to fit the computational domain. Distribution of physical field characteristic functions within and computational domain Distribution of physical field characteristic functions within , These are trainable parameters representing the feature values. These are the trainable parameters for the fully connected layer modules. and These are the diffusion coefficients of the materials on both sides, Training points used to contact the boundary.

7. The improved PINNs-based simulation method of neutron diffusion according to claim 2, characterized in that, The vacuum extrapolation boundary loss is specifically as follows: when the vacuum boundary is located in the calculation domain 1, the vacuum extrapolation boundary loss is wherein: is the training point after the vacuum boundary is extrapolated, and a calculation formula thereof is , and , represents the diffusion coefficient of the material at the extrapolation boundary.

8. The improved PINNs-based simulation method of neutron diffusion according to claim 2, characterized in that, The total loss function is given by where: is the corresponding weight for each individual loss, is the boundary condition loss, is the fixed point loss, is the PDE loss, , are the contact boundary losses, is the vacuum extrapolation boundary loss.

9. A neutron diffusion simulation system implementing the method of any of claims 1-8, characterized by The method comprises the following steps: The full connection layer module, the automatic differentiation module, the loss function module, the fixed point transformation module, the calculation domain splicing module and the vacuum extrapolation boundary module, wherein: the full connection layer module reads the training points according to the weights, the bias and the additional trainable parameters representing the predicted values of the characteristic values, solves the neutron diffusion equation to obtain a trial function, and creates a boundary condition loss; the fixed point transformation module performs fixed point transformation on the trial function to obtain a trial function after fixed point transformation constraint and create a fixed point loss; the automatic differentiation module performs automatic differentiation on the trial function updated by the fixed point transformation module according to the time, space coordinates of the training points, and creates a PDE loss; the calculation domain splicing module divides the trial function after fixed point transformation constraint according to the calculation domains of different materials of the physical model, reads the additional set of contact boundary training points, and creates a contact boundary loss; the vacuum extrapolation boundary module performs boundary extrapolation according to the training points of the vacuum boundary to create a vacuum extrapolation boundary loss; and the loss function module constructs a total loss function according to the PDE loss, the boundary condition loss, the fixed point loss, the contact boundary loss and the vacuum extrapolation boundary loss, which is used for training the improved PINNs.

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