An optical element surface shape simulation method based on a physical information neural network

Abstract

The application provides an optical element surface simulation method based on a physical information neural network, relates to the technical field of element design and simulation, and comprises the following steps: constructing a feedforward neural network; generating a deformation control equation model library; constructing a three-dimensional simulation database; forming a machine learning parameter library; training a PINN deformation simulation model in combination with the deformation control equation model library, the three-dimensional simulation database and the machine learning parameter library; screening out a target PINN deformation simulation model meeting preset conditions; inputting design parameters, support structure parameters and given external load parameters of different combinations into the target PINN model to obtain key performance prediction results of the optical element; the application can not only quickly evaluate the key performance of the element under various parameter combinations, shorten the design iteration cycle and reduce the trial and error cost, but also provides direct data support for support structure optimization design, and effectively solves the bottleneck problem of the traditional method in restricting the research and development efficiency of a high-precision optical system.

Description

A method for simulating the surface profile of optical components based on physical information neural networks Technical Field

[0001] This invention relates to the field of optical component design and simulation technology, and in particular to an optical component surface simulation method based on physical information neural networks. Background Technology

[0002] The surface accuracy of optical components (such as mirrors and lenses) is a key indicator determining the performance of an optical system. In practical applications, optical components are inevitably affected by their own weight, the assembly constraints of specific support structures (frames, tooling), and external environmental loads (such as thermal loads caused by temperature gradients and acceleration loads caused by platform vibrations), resulting in elastic deformation and a decrease in surface accuracy. Therefore, accurate and efficient simulation prediction of surface deformation under complex operating conditions is a crucial technical step in ensuring component performance during the design and manufacturing stages of optical systems.

[0003] Currently, surface modeling of optical components primarily relies on traditional numerical methods, such as the Finite Element Method (FEM). When dealing with deformation problems of optical components, the FEM requires fine meshing of the component, supporting structure, and load-contact areas. However, for complex geometries, multi-point support constraints, and especially for complex multi-field load problems involving thermo-mechanical coupling and transient dynamics, mesh generation is extremely difficult, and generating high-quality meshes is often very time-consuming. Furthermore, solving large sparse matrices in high-dimensional, complex FEM models requires enormous computational resources and lengthy solution times, making it difficult to meet the efficiency requirements of design iteration and rapid optimization.

[0004] Furthermore, the traditional finite element method has limitations in solving inverse problems. For example, it is difficult to efficiently and accurately deduce the support structure parameters or unknown load distributions that cause deformation based on actual measured surface error data. This limitation restricts designers' ability to quickly evaluate and optimize support schemes and tooling parameters, resulting in long design cycles and high trial-and-error costs for the support structure optimization of optical components. This has become a significant bottleneck restricting the development efficiency of high-precision optical systems. Summary of the Invention

[0005] To address the aforementioned technical problems, the technical solution adopted by this invention is as follows:

[0006] According to a first aspect of this application, a method for simulating the surface profile of an optical element based on a physical information neural network is provided, the method comprising the following steps:

[0007] S100, based on the three-dimensional model library corresponding to the optical element, constructs a feedforward neural network using the physical information neural network method; the input of the feedforward neural network is spatial coordinates, time and load parameters, and the output is a three-dimensional displacement field; the three-dimensional model library is obtained from the parameter library corresponding to the optical element; the parameter library includes the design parameters of the optical element, the support structure parameters, the material property parameters and the external load parameters.

[0008] S200, the loss function of the physical field differential equation describing the deformation of the optical element is embedded in the neural network as a physical constraint, and the corresponding PINN deformation simulation model is constructed according to different support structure parameters to generate a deformation control equation model library.

[0009] S300, based on the clearly defined boundaries of the 3D model library, defines complex support conditions and load conditions as boundary conditions, constructs the PINN total loss function which includes physical loss, boundary loss and data loss, and uses the intra-domain and boundary point-scattering methods to generate spatial coordinate point-scattering data for training the PINN deformation simulation model, and constructs a 3D simulation database.

[0010] S400 calls the deformation control equation model library, PINN total loss function and three-dimensional simulation database, uses physical information neural network to perform multi-condition generalization training, and establishes a correlation mapping database between various parameters in the parameter library and the surface accuracy index of optical components.

[0011] S500, based on the aforementioned associated mapping database as the initialization basis, combined with the deformation control equation model library and the three-dimensional simulation database, the PINN deformation simulation model is refined and its physical constraint is verified, and target PINN deformation simulation models that meet the preset conditions are selected.

[0012] S600 inputs different combinations of design parameters, support structure parameters, and given external load parameters into the target PINN model to obtain the key performance prediction results of the optical element.

[0013] The present invention has at least the following beneficial effects:

[0014] The optical component surface simulation method based on Physical Information Neural Network (PINN) of this invention effectively solves the problems of difficult mesh generation, high computational resource consumption, and low solution efficiency in the traditional finite element method (FEM) for optical component surface simulation. It also overcomes the limitations of traditional methods in solving inverse problems. Specifically, by constructing a complete parameter library containing design parameters, support structure parameters, material property parameters, and external load parameters, and by clearly defining the physical computation domain and boundaries based on a 3D model library, a comprehensive and accurate basic input for simulation is provided. Furthermore, by constructing a feedforward neural network using a Physical Information Neural Network (PINN), the physical field differential equations are embedded into a loss function to form physical constraints, ensuring the physical consistency of the simulation results. The method offers rationality and interpretability, and eliminates the need for complex mesh preprocessing, significantly improving solution efficiency in transient or nonlinear deformation scenarios. By constructing a 3D simulation database using scattered data and establishing a mapping relationship between parameters and surface accuracy indicators through model training, the resulting machine learning parameter library supports rapid correlation between different parameter combinations and simulation results. Combined with the selection of target PINN models, it achieves efficient and accurate surface prediction of optical components under complex working conditions. The overall approach not only quickly evaluates the key performance of components under various parameter combinations, shortening the design iteration cycle and reducing trial and error costs, but also provides direct data support for structural optimization design, effectively solving the bottleneck problem that traditional methods restrict the R&D efficiency of high-precision optical systems. Attached Figure Description

[0015] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0016] Figure 1 is a flowchart of the optical element surface simulation method based on physical information neural network provided in an embodiment of the present invention;

[0017] Figure 2 is a comparison and overall flowchart of the optical element surface simulation method based on physical information neural network provided in this embodiment of the invention with existing methods;

[0018] Figure 3 is a schematic diagram of the PINN network provided in an embodiment of the present invention. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] It should be noted that, based on this disclosure, those skilled in the art will understand that one aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects set forth herein can be used to implement the device and / or practice the method. Furthermore, this device and / or practice the method can be implemented using other structures and / or functionalities besides one or more of the aspects set forth herein.

[0021] The following will introduce a method for simulating the surface shape of optical components based on a physical information neural network, with reference to the flowchart of the method shown in Figure 1.

[0022] The optical component surface simulation method based on physical information neural networks may include the following steps:

[0023] S100, based on the three-dimensional model library corresponding to the optical element, constructs a feedforward neural network using the physical information neural network method; the input of the feedforward neural network is spatial coordinates, time and load parameters, and the output is a three-dimensional displacement field; the three-dimensional model library is obtained from the parameter library corresponding to the optical element; the parameter library includes the design parameters of the optical element, the support structure parameters, the material property parameters and the external load parameters.

[0024] In this embodiment, the method used is compared with existing methods, and the overall process is shown in Figure 2. Before establishing the parameter library, the parameters of the following optical components need to be obtained:

[0025] Design parameters can be the radius R (example: 0.5m~2m), thickness T (example: 5mm~20mm), and radius of curvature C (example: 5m~20m, for plane mirrors, C is taken as infinity).

[0026] Support structure parameters: Support point locations (example: 3-point support corresponds to coordinates (0°, 0°), (120°, 0°), (240°, 0°), 9-point support adds intermediate layer coordinates on the basis of 3 points), Support tooling material parameters (elastic modulus E: 200GPa~210GPa, Poisson's ratio μ: 0.27~0.30, thermal conductivity k: 150W / (m•K)~180W / (m•K));

[0027] Material property parameters: the elastic modulus E of the optical element itself (example: fused silica E=72GPa), Poisson's ratio μ (example: 0.17), thermal conductivity k (example: 1.38W / (m•K)), density ρ (example: 2200kg / m³), and specific heat capacity at constant pressure c. p (Example: 741 J / (kg•K));

[0028] External load parameters: thermal load gradient (example: -50℃ / m~50℃ / m), acceleration load a (example: 0g~10g, g=9.8m / s²).

[0029] After obtaining the above parameters, Latin hypercube sampling (LHS) can be used to generate 300-500 sets of samples within the above range to ensure uniform parameter distribution and avoid duplication and redundancy. It should be noted that those skilled in the art can use existing LHS methods to generate 300-500 sets of samples based on the above parameters according to actual needs, which will not be elaborated here.

[0030] Using SolidWorks or Zemax parametric modeling software, import the parameter set generated by LHS to automatically generate 3D solid models of optical components (such as mirror surfaces), supporting frames, and tooling, and define the physical computational domain Ω (example: the computational domain of the mirror is a cylinder with a radius of 0.5m and a thickness of 10mm) and its boundaries. (Mirror surface, mirror frame contact surface, tooling support surface); categorized and stored in the format of "parameter group number-model file" to form a 3D model library, supporting subsequent indexing by parameter.

[0031] Feedforward neural networks can be constructed using the following method:

[0032] The network structure is set as follows: Input layer (6 neurons, corresponding to spatial coordinates x / y / z (range: x∈[-0.5m,0.5m], y∈[-0.5m,0.5m], z∈[0,0.01m]), time t (example: 0s~10s), load parameters) → Hidden layer (9 layers, 80 neurons per layer) → Output layer (3 neurons, corresponding to the three-dimensional displacement field u) x u y u_z (unit: μm).

[0033] Initialization configuration: The activation function is the Tanh function, and the network weights and biases are initialized using the He method, implemented through the torch.nn.init.kaiming_uniform of the PyTorch 1.10+ framework;

[0034] Network compilation: Set up the network forward propagation process to ensure that the input parameters are standardized (x'=(x-μ) / σ, where μ is the mean of the parameters and σ is the standard deviation) before being input into the network.

[0035] This step ensures that the simulation input covers all dimensions of optical element design, support, materials, and loads by constructing a complete and quantified parameter library, avoiding the result deviations caused by missing parameters in traditional simulations. The 3D model library is generated based on parametric modeling, clarifying the physical computation domain and boundaries, providing accurate geometric basis for subsequent point casting and boundary condition definition, and solving the problem of ambiguous computation domain in traditional methods. The structure and initialization parameters of the feedforward neural network are clear and can be directly reproduced through mainstream deep learning frameworks. Its input and output are highly matched with the physical process of optical element deformation, laying the foundation for subsequent physical constraint embedding. Overall, the pre-simulation work is standardized and reusable, significantly reducing the implementation threshold of subsequent steps.

[0036] S200, the loss function of the physical field differential equation describing the deformation of the optical element is embedded in the neural network as a physical constraint, and a corresponding PINN deformation simulation model is constructed according to different support structure parameters to generate a deformation control equation model library.

[0037] Furthermore, step S200 includes the following steps:

[0038] S210, Set up and initialize the structure of the feedforward neural network; wherein the feedforward neural network has 6-12 layers and 60-100 neurons per layer, select Tanh or Sigmoid as the activation function, use the He initialization method to initialize the network weights and biases, and construct the network forward propagation process.

[0039] In this embodiment, the structure of the PINN deformation simulation model is shown in Figure 3. The network size can be adapted according to the aperture of the optical element. For small-aperture elements (radius ≤ 0.5m, such as small lenses), 6 hidden layers are used, with 60 neurons per layer; for medium-aperture elements (0.5m < radius < 1m, such as industrial mirrors), 9 hidden layers are used, with 80 neurons per layer; and for large-aperture elements (radius ≥ 1m, such as aerospace optical mirrors), 12 hidden layers are used, with 100 neurons per layer. The input layer is fixed at 6 neurons, corresponding to spatial coordinates (x, y, z), time t, thermal load gradient, and acceleration load; the output layer is fixed at 3 neurons, corresponding to the three-dimensional displacement field (u). x u y 、u_z).

[0040] The Tanh function is preferred as the activation function, as it is suitable for large-aperture, high-load conditions (such as impact loads during aerospace launches). Its output range is symmetrical and its gradient is stable, which is beneficial for calculating higher-order partial derivatives. For small-aperture, low-load conditions (such as lenses in laboratory environments), the Sigmoid function can be chosen to reduce computational complexity.

[0041] Weights and biases are initialized using the He initialization method. Specifically, for the weights of each linear layer in the network, they are determined by sampling from a uniform distribution with a mean of 0 and a variance of 2 / number of input neurons. Biases are uniformly initialized to 0. For example, the sampling range for the weights from the input layer (6 neurons) to the hidden layer (80 neurons) is [-(2 / 6)]. 1 / 2 (2 / 6) 1 / 2 ], approximately [-0.577, 0.577].

[0042] Forward propagation process construction: The signal transmission flow is "input layer → hidden layer → output layer". The input parameters are standardized (x'=(x-μ) / σ, μ is the mean of the parameters, and σ is the standard deviation) and then input into the input layer. The input of each hidden layer is linearly transformed (weight × input + bias) and then transformed into the output through the activation function. The output layer has no activation function and directly outputs the original values ​​of the three-dimensional displacement field.

[0043] Example code is as follows:

[0044] import torch

[0045] import torch.nn as nn

[0046] class PINN_Net(nn.Module):

[0047] def__init__(self,layers=9,neurons_per_layer=80):

[0048] super(PINN_Net,self).__init__()

[0049] #Input layer: 6-dimensional (spatial coordinates x / y / z + time t + thermal load gradient + acceleration load)

[0050] self.input_layer=nn.Linear(6,neurons_per_layer)

[0051] #Hidden layers: 6-12 layers (default 9 layers), 60-100 neurons per layer (default 80 neurons)

[0052] self.hidden_layers=nn.ModuleList()

[0053] for_in range(layers-2):

[0054] self.hidden_layers.append(nn.Linear(neurons_per_layer, neurons_per_layer))

[0055] #Output layer: 3D (3D displacement field ux / uy / uz)

[0056] self.output_layer=nn.Linear(neurons_per_layer,3)

[0057] #Activation function selection: Tanh (default) or Sigmoid

[0058] self.activation=nn.Tanh() # Replace with nn.Sigmoid() to toggle the activation function.

[0059] def forward(self,x):

[0060] #Forward Propagation Process

[0061] out=self.activation(self.input_layer(x))

[0062] for layer in self.hidden_layers:

[0063] out=self.activation(layer(out))

[0064] out = self.output_layer(out)

[0065] return out.

[0066] This step avoids subjectivity in network structure selection by adhering to the rule of "aperture-fitting network size," ensuring the relevance of simulations for optical components of different sizes. The He initialization method, combined with the Tanh / Sigmoid activation function, effectively alleviates the gradient vanishing or exploding problem in deep network training, improving the stability of training convergence. The standardized forward propagation process ensures the consistency of input parameters, laying a unified foundation for subsequent physical constraint embedding and residual calculation. Overall, it achieves reproducibility and reusability in network construction, balancing the computational efficiency of small-aperture components with the fitting accuracy of large-aperture components.

[0067] S211: Introduce the elasticity equilibrium differential equation and the heat conduction differential equation into the physical loss function; among them, the stress tensor in the elasticity equilibrium differential equation is related to the three-dimensional displacement field through the thermoelastic constitutive relation.

[0068] Furthermore, in the elasticity equilibrium differential equation, the volume force f = ρa; where ρ is the density of the optical element and a is the acceleration load in the external load parameters; in the heat conduction differential equation, the internal heat source Q is defaulted to 0, and if the internal heat source is considered, Q is included in the external load parameter library.

[0069] In this embodiment, all physical parameters are selected based on actual material properties. Taking fused silica optical element + aluminum alloy support fixture as an example: the density of fused silica ρ = 2200 kg / m³, and the specific heat capacity at constant pressure c p =741 J / (kg•K), thermal conductivity k=1.38 W / (m•K), elastic modulus E=72 GPa, Poisson's ratio μ=0.17, coefficient of thermal expansion α=0.55×10 -6 / K; The elastic modulus of aluminum alloy E=200GPa, Poisson's ratio μ=0.30; The internal heat source Q is set to 0 by default (in the scenario without an internal heat source), and the volume force f=ρa (a is the acceleration load).

[0070] The equilibrium differential equation of elasticity is: The differential equation for heat conduction is: Among them, the stress tensor σ is obtained through the thermoelastic constitutive relation. In relation to the three-dimensional displacement field, the specific derivation is as follows: First, the strain tensor is obtained from the geometric equations. (where u is the three-dimensional displacement field), and the thermal strain tensor is obtained from the temperature change ΔT (ΔT = thermal load gradient × z coordinate). Finally, the strain difference is expressed using Lamé constants λ=(Eμ) / [(1+μ)(1-2μ)] and μ=E / [2(1+μ)]. It is converted into the stress tensor σ.

[0071] The physical loss L_Physics is the mean square sum of the residuals from the two equations. The calculation logic is as follows: first, calculate the residuals of the elasticity equilibrium equations separately ( The actual calculated value) and the residual of the heat conduction equation ( The actual calculated value), then the average of the two residuals is calculated separately and then summed, that is, L_Physics = mean (elasticity mechanical residual). 2 ) + mean (thermal conduction residual) 2 ).

[0072] This step directly transforms the core physical laws of thermo-mechanical coupling into constraints of the loss function, fundamentally ensuring the physical rationality of the simulation results and avoiding the defects of traditional data-driven models that "only fit the data but violate objective laws" (such as contradictory results like deformation under no load or shrinkage in high-temperature regions). The complete introduction of thermoelastic constitutive relations accurately establishes the correlation between the temperature field, strain field, stress field, and displacement field, solving the problem of fragmented thermo-mechanical coupling analysis in traditional simulations and accurately simulating the real deformation of optical components under superimposed loads. All physical parameters are set based on actual material properties, ensuring the engineering practicality of the loss function and providing a realistic optimization target for subsequent model training.

[0073] S212: Obtain the first and second order partial derivatives of the feedforward neural network output with respect to the input spatial coordinates, substitute them into the physical field differential equation to calculate the residual, so as to ensure that the neural network output meets the physical constraints.

[0074] This system utilizes the automatic differentiation capabilities of deep learning frameworks (such as PyTorch and TensorFlow), eliminating the need for manual derivation of derivative formulas. The steps are as follows: First, set the network input (spatial coordinates x / y / z) to "differentiable" mode; after the network outputs a 3D displacement field, call the framework's gradient calculation interface to sequentially calculate the first-order partial derivatives of the displacement field with respect to the spatial coordinates (e.g., u...). x Partial derivative with respect to x u y Partial derivative with respect to y There are a total of 6 independent first-order partial derivatives); based on the first-order partial derivative results, the gradient calculation interface is called again to calculate the second-order partial derivatives (e.g., ...). (etc., to meet the requirements of the physical equations).

[0075] Verification was conducted using simple operating conditions with known analytical solutions, such as a fused silica element with a uniform thermal load gradient of 10℃ / m and no acceleration load, to determine its thermal strain. =α×10=5.5×10 -6 The corresponding first-order partial derivative It should be equal to this value, and the deviation between the automatic differential calculation result and the analytical solution should be ≤1e-8 to ensure the accuracy of the derivative calculation.

[0076] Substitute the first and second order partial derivatives into the elasticity equilibrium differential equation and the heat conduction differential equation, and calculate the equation residual at each point (residual = calculated value on the left side of the equation - theoretical value on the right side of the equation, e.g., elasticity residual = ...). In the total loss function, the weight of physical loss is set to 0.7, the weight of boundary loss to 0.2, and the weight of data loss to 0.1. During training, physical residuals are minimized first to ensure that the network output meets physical constraints.

[0077] This step utilizes automatic differentiation technology to replace the manual differentiation and mesh discretization of the traditional finite element method, avoiding the tedious calculations and errors of manually deriving higher-order derivatives, and significantly improving the efficiency and accuracy of residual calculation. The complete acquisition of first- and second-order partial derivatives ensures comprehensive constraints on the physical field differential equations, solving the problem of insufficient physical constraints caused by the traditional PINN only considering the first-order derivative. By strengthening the optimization priority of physical residuals through weight allocation, even when training data is insufficient or noisy, the network output can be guaranteed to conform to the objective laws of elasticity and heat conduction, significantly improving the reliability and interpretability of simulation results.

[0078] S213: For different support point locations, support tooling material parameters, and other support structure parameters, instantiate the corresponding differential equation models to obtain the corresponding PINN deformation simulation models, thereby generating a deformation control equation model library.

[0079] The support structure parameters are divided into two categories: first, the support point positions, including 3-point support (coordinates (0°,0°), (120°,0°), (240°,0°)), 9-point support (adding 6 intermediate points such as (60°,50mm), (180°,50mm) on the basis of 3 points), and support point offsets (±5mm, ±10mm); second, the support tooling materials, including aluminum alloy, titanium alloy, and stainless steel (corresponding to elastic moduli of 200GPa, 110GPa, and 210GPa, and Poisson's ratios of 0.30, 0.34, and 0.27).

[0080] Using the basic network constructed in S210 and the physical loss function in S211 as templates, only the "physical parameters related to the support" are modified for different support structure parameters, without changing the core physical equations and network structure: if the support tooling material is changed, only the tooling elastic modulus E and Poisson's ratio μ in the physical loss function are updated; if the support point position is changed, only the "spatial coordinates of the support constraint" in the subsequent boundary conditions are adjusted, while the differential equation form remains unchanged. For example, when instantiating the "3-point support + titanium alloy tooling" model, only the tooling material parameters are updated to E=110GPa and μ=0.34 for titanium alloy, while the other parameters remain unchanged.

[0081] Each instantiated PINN deformation simulation model is stored separately, with the naming convention being "Number of Support Points - Tooling Material - Support Point Offset" (e.g., "3-Point Support - Aluminum Alloy - 0mm.pth" "9-Point Support - Titanium Alloy - 5mm.pth"). They are stored in folders according to the hierarchical structure of "Number of Support Points → Tooling Material → Offset", for example, "Deformation Control Equation Model Library / 3-Point Support / Aluminum Alloy / 0mm.pth", which facilitates quick indexing and retrieval by working condition.

[0082] This step, through the "template-based instantiation" rule, avoids the repetitive work of redesigning networks and physical equations for each support condition, significantly improving the efficiency of multi-condition modeling. The hierarchical storage and standardized naming of the model library enable rapid switching and management of models with different support schemes, solving the tedious problem of remodeling required for switching between multiple conditions in traditional simulations. The instantiation process only modifies support-related parameters, keeping the core physical equations consistent, ensuring the physical consistency of simulation results for different support schemes. This facilitates direct comparison of the impact of different support structures on the surface deformation of optical elements, providing efficient and unified model support for support scheme optimization.

[0083] Furthermore, the PINN deformation simulation model described in step S200 is a dual-network collaborative architecture of a physical dominant network and a data correction network, specifically implemented by the following steps:

[0084] S220, construct a physical dominant network, input spatial coordinates, time, load parameters and optical element material parameters, output a preliminary three-dimensional displacement field, set the loss function of the physical dominant network to be mainly physical loss L_Physics, the initial proportion of physical loss weight is ≥α1, the physical loss is calculated by the residuals of the elasticity equilibrium differential equation and the heat conduction differential equation; α1 is the first preset proportion threshold.

[0085] In this embodiment, the feedforward neural network architecture of S210 is adopted, and the scale is adapted according to the aperture of the optical element. The input layer adds optical element material parameters (elastic modulus E, Poisson's ratio μ, thermal conductivity k, density ρ, and specific heat capacity at constant pressure c). p The total input dimension is 12 (spatial coordinates x / y / z + time t + thermal load gradient + acceleration load + 6 material parameters); the output layer is still 3-dimensional, corresponding to the preliminary three-dimensional displacement field (u x 1 u y 1 u_z 1 The activation function is preferably Tanh, and the weights and biases are initialized using He (the weights follow a uniform distribution with a mean of 0 and a variance of 2 / number of input neurons, and the biases are initialized to 0).

[0086] The first preset percentage threshold α1 is fixed at 70%, that is, the initial percentage of physical loss weight is ≥70%, the percentage of boundary loss weight is ≤20%, and the percentage of data loss weight is ≤10% (only as an auxiliary constraint).

[0087] Physical loss L_Physics calculation: can completely follow the core logic of S211, based on the elasticity equilibrium differential equation ( ) and the differential equation of heat conduction Calculate the residuals, L_Physics = mean (elasticity residuals)2 ) + mean (thermal conduction residual) 2 For example, when a fused silica element is subjected to an acceleration load of 5g, the stress and displacement are correlated through thermoelastic constitutive relations, the equation residuals at each point are calculated, and the summation and average value is taken as the physical loss.

[0088] Initial training conditions: Adam optimizer is used, learning rate is 1e-4, batch size is 2048, and training epochs are 1000; termination condition is physical loss ≤1e-5 or loss decrease ≤1e-8 for 200 consecutive generations, to ensure that the network first masters the core physical laws.

[0089] This step strengthens physical constraints by fixing α1=70%, ensuring that the physics-driven network prioritizes fitting objective laws such as elasticity and heat conduction. This fundamentally avoids the defect of traditional single networks that "overfit data but violate physical logic" (such as deformation when there is no load). The input layer incorporates complete material parameters, enabling the network to adapt to optical components of different materials and improving the model's versatility. The network structure and basic training conditions are clear and highly reproducible. Furthermore, the training results (preliminary displacement field) provide a "physically reasonable" basis for subsequent data correction, preventing the correction network from getting bogged down in meaningless accuracy optimization.

[0090] S221, Construct a data correction network. Take the preliminary three-dimensional displacement field output by the physical dominant network as input, combine it with the finite element high-precision simulation data or experimental measurement data of the effective optical area of ​​the optical element, and output the final three-dimensional displacement field. Set the loss function of the data correction network to be mainly based on the data loss L_Data. The initial proportion of the data loss weight is ≥ α2. The correction loss is calculated only in the effective optical area. α2 is the second preset proportion threshold. α1 < α2.

[0091] The data correction network has a network size half that of the physics-dominated network (small-aperture 3 layers × 60 neurons, medium-aperture 5 layers × 80 neurons, large-aperture 6 layers × 100 neurons), reducing computational complexity. The input layer is 4-dimensional: the preliminary three-dimensional displacement field (u) output by the physics-dominated network. x 1 u y 1 u_z 1 + Effective optical region identifier (0 or 1, 1 indicates that the point belongs to the effective optical region); the output layer is 3D, corresponding to the final three-dimensional displacement field (u x 2 u y 2 u_z 2 The activation function is Tanh, and the weights and biases are initialized using He.

[0092] The second preset percentage threshold α2 is fixed at 80%, that is, the initial percentage of data loss weight is ≥80%, the weight of physical loss is ≤10%, the weight of boundary loss is ≤10%, and the accuracy correction of the effective optical area is focused.

[0093] Effective optical area division: Based on the 3D model library, the effective optical area of ​​the reflector is the core area of ​​the mirror surface (radius = total radius of the component × 90%, such as a reflector with a total radius of 0.5m, the effective area radius is 0.45m); the effective optical area of ​​the lens is the light-transmitting area (excluding the part obstructed by the lens frame), and a 1mm wide transition zone is set at the edge and where the curvature changes abruptly.

[0094] Finite element high-precision simulation data should be calculated using a fine mesh with a mesh size ≤ 0.1 mm (such as ANSYS software, with the solution accuracy set to the highest level); experimental measured data are obtained using a Zygo GPI XPS interferometer with a measurement accuracy ≤ 0.01 μm and a sampling density ≥ 100 points / mm², and only data from the effective optical area are retained for training.

[0095] Data loss L_Data calculation: Calculated only within the effective optical area, L_Data = mean[(u x 2 -u x _ref) 2 +(u y 2 -u y _ref) 2 +(u_z 2 -u_z_ref) 2 ]; where u x _ref、u y _ref and u_z_ref are high-precision reference displacement fields for finite element methods or experiments.

[0096] This step, by setting a weight of α2=80%, forces the data correction network to focus on the accuracy optimization of the effective optical region, solving the pain point of traditional PINN's "balanced accuracy across the entire domain but insufficient accuracy in key areas", and significantly improving the simulation accuracy of the core working area of ​​the optical element. The strict requirements of high-precision data for the quantitative division of the effective optical region ensure the pertinence and reliability of the correction. The network size is designed to be half that of the physical dominant network, which controls the computational cost while ensuring accuracy, and achieves a balance of "high local accuracy + high global efficiency".

[0097] S222, the physical dominant network and the data correction network are trained alternately. The training results of the physical dominant network are used as the initial input of the data correction network. The correction error of the data correction network is fed back to the physical dominant network through the backpropagation mechanism. The physical loss weight of the physical dominant network is dynamically adjusted to achieve a balance between global physical rationality and local high precision in the effective optical area.

[0098] Alternating training sequence and rounds: The process follows a sequence of "physically-driven network training → data-corrected network training → error feedback adjustment → repetition," alternating for a total of 2-3 rounds.

[0099] First round: Train the physical dominant network to convergence according to the conditions of S220 to obtain the initial displacement field; input the displacement field and high-precision data into the data correction network, and train it according to the conditions of S221 until the data loss is ≤1e-6;

[0100] Second round: Extract data to correct the network's correction error (u_z) 2 -u_z 1 ), calculate the mean error, if the mean error is > 0.02μm (effective optical area), then reduce the physical loss weight of the physical dominant network by 5%-10% (e.g., from 70% to 65%), retrain the physical dominant network; then input the new initial displacement field data to correct the network and retrain it;

[0101] Third round: If the mean correction error is still >0.01μm, repeat the weight adjustment and training of the second round until the mean error is ≤0.01μm or the weight is reduced to 50% (the minimum threshold to ensure that the physical constraints do not fail).

[0102] The correction error of the data correction network is backpropagated through gradients and directly affects the loss function of the physical dominant network, adjusting its weight update direction. If the correction error at a certain point in the effective optical region is large, the physical loss weight of the physical dominant network at that point will be reduced accordingly, making the network output more consistent with high-precision data.

[0103] After alternating training of the two networks, the physical loss of the physical dominant network is ≤1e-5, the data loss of the data correction network is ≤1e-6, and the RMS error of the effective optical region is ≤0.03μm. At this point, training is stopped, and the final PINN deformation simulation model is output.

[0104] The alternating training mechanism in this step breaks the deadlock of traditional single networks where "physical constraints and data fitting are difficult to balance." By dynamically adjusting the physical loss weights, it achieves a precise balance between "global physical rationality" and "local high precision in the effective optical region." Backpropagation feedback enables the physical-dominated network to specifically optimize regions with large errors, avoiding blind adjustments. Clear alternation rounds and convergence conditions ensure the reproducibility of the training process and the stability of the results. The final model not only conforms to objective physical laws but also meets the high precision requirements of the core working area of ​​optical components, significantly improving the engineering practical value of the simulation results.

[0105] Furthermore, the load parameters mentioned in step S100 include time-varying loads, and the PINN deformation simulation model in step S200 integrates a time-varying load adaptive modeling and uncertainty quantification module, specifically implemented through the following steps:

[0106] S230, a load time-varying feature extraction module is added to the input layer of the PINN deformation simulation model. An attention mechanism is used to capture the time evolution features of thermal load and acceleration load. The time evolution features include the peak moment of the impact load and the period of temperature fluctuation. The time step and activation function weight of the physical dominant network are dynamically adjusted according to the feature extraction results to adapt to transient and periodic time-varying conditions.

[0107] In this embodiment, the load time-varying feature extraction module is connected in series between the input layer of the PINN deformation simulation model and the hidden layer of the physical dominant network. The module includes a "time series encoding unit" and an "attention weight calculation unit". The input is load data in time series form (curve of thermal load gradient change with time, curve of acceleration load change with time), and the output is a weighted time-varying feature vector.

[0108] The operation process of the attention mechanism is as follows:

[0109] Time series encoding: The payload data is discretized into a time series according to time intervals (initially set to 0.01s). The payload value at each time step and the corresponding timestamp form a feature vector (e.g., payload value q0 at time t=0s, q1 at t=0.01s, forming the sequence [q0,t0;q1,t1;...;qn,tn]).

[0110] Attention weight calculation: An additive attention mechanism is used to calculate the attention weight of the feature vector at each time step. For impact loads (such as the 5g impact during a space launch), the weights are tilted towards the time steps near the peak load (the weights of the three time steps before and after the peak are amplified by 2 times). For periodic temperature fluctuations (such as ±20℃ fluctuations with a period of 10s), the weights are dynamically allocated according to the periodic pattern (the weights of the key nodes in each period <heating start, peak, cooling end> ​​are increased by 1.5 times).

[0111] Temporal evolution feature extraction: The core features are obtained by weighted summation: the peak moment of the impact load (the t value corresponding to the time step with the highest weight) and the period of temperature fluctuation (the time difference between two adjacent peak moments).

[0112] Dynamic adjustment rules:

[0113] Time step adjustment: If an impact load is extracted (within 0.1s before and after the peak moment), reduce the network's time step from 0.01s to 0.001s to improve the accuracy of transient response capture; if it is a periodic load (period T≥1s), set the time step to T / 50 (e.g., when T=10s, the time step = 0.2s) to balance accuracy and efficiency.

[0114] Activation function weight adjustment: For regions with a load change rate ≥10℃ / (m•s) or ≥1g / s, increase the Tanh activation function weights of the first 3 hidden layers of the physically dominant network by 10%-20% (the weight adjustment range is 1.1-1.2 times the original weights) to enhance the network's ability to fit load abrupt changes.

[0115] Example: When the input acceleration load is an impact load that increases from 0g to 5g (peak) in 0-0.1s and decreases from 5g to 0g in 0.1-0.2s, the module extracts the peak time t=0.1s, reduces the time step to 0.001s, and adjusts the weight of the activation function of the first 3 layers from the original 0.8 to 0.96 to adapt to the transient impact response.

[0116] This step accurately captures the core evolution characteristics of time-varying loads through an attention mechanism, solving the problem of poor adaptability and delayed response of traditional PINN for transient and periodic loads. The dynamic adjustment of the time step and activation function weights achieves a balance between high precision in load change regions and high efficiency in stable regions, avoiding the surge in computational load caused by a fixed small step size across the entire domain. The seamless connection between the module and the physical-dominated network ensures that time-varying features can directly guide network parameter optimization, significantly improving the accuracy and timeliness of optical component deformation simulation under dynamic conditions.

[0117] S231 employs a Bayesian physical information neural network architecture to quantify uncertainty. The weights and biases of the physical dominant network are defined as Gaussian probability distributions. Several sets of model parameter samples are generated using the Markov chain Monte Carlo sampling method, and the 95% confidence interval of the surface accuracy index is output to quantify the impact of material parameter fluctuations and measurement errors on the simulation results.

[0118] Based on the physical dominant network, the weights and biases of all linear layers are defined as Gaussian probability distributions—the weight distribution is N(μ_w,σ_w). 2 ), where μ_w is the weight value initialized by He in S210, σ_w2 =0.01 (initial variance); the biased distribution is N(0,0.001). 2 This ensures the rationality of the distribution.

[0119] The Markov chain Monte Carlo (MCMC) sampling operation is as follows:

[0120] Sampling algorithm selection: The No-U-Turn Sampler (NUTS) algorithm is adopted to adapt to efficient sampling in high-dimensional parameter spaces;

[0121] Sampling parameter settings: Set 4 independent sampling chains, each chain has a combustion period (preheating period) of 1000 steps, the formal sampling steps are 2000 steps, the sampling interval is 10 steps (to avoid sample correlation), and the total number of samples = 4 × (2000 / 10) = 800 groups;

[0122] Sample screening: Remove samples within the combustion period, perform consistency tests on the remaining samples (R-hat statistic ≤ 1.05, to ensure sampling convergence), and retain qualified samples to form the model parameter sample set.

[0123] Confidence intervals and uncertainty quantification:

[0124] Surface accuracy index calculation: Substitute 800 sets of model parameter samples into the physical dominant network to output 800 sets of three-dimensional displacement fields, and calculate the effective optical region PV value and RMS value for each set.

[0125] 95% confidence interval calculation: Sort the sample set of PV values ​​and RMS values ​​from smallest to largest, and take the 2.5% quantile and 97.5% quantile to form the 95% confidence interval (e.g., PV value: 0.08-0.11μm, RMS value: 0.02-0.03μm).

[0126] Quantification objects: Identify the sources of uncertainty in quantification, including material parameter fluctuations (such as ±5% fluctuation in the elastic modulus of fused silica), load measurement errors (such as ±3% error in acceleration load), and the impact of point data noise (±1e-7μm) on simulation results.

[0127] This step transforms network parameters from fixed values ​​to a probability distribution using a Bayesian architecture, overcoming the limitations of traditional PINN which "only outputs single-point predictions and has no error boundaries." The standardized MCMC sampling process ensures the reliability and convergence of the samples, and the 95% confidence interval intuitively quantifies the impact of various errors on the simulation results, allowing designers to clearly understand the reliable range of the results. Uncertainty quantification provides a clear target for subsequent error compensation, avoiding the engineering risk of "simulation results appearing accurate but actual deviations being too large," and significantly improving the credibility and engineering practicality of the simulation results.

[0128] S232, determine whether the confidence interval exceeds the preset allowable range. If it does, automatically increase the weight ratio of physical loss under the corresponding load condition, supplement the scattered data under the corresponding load condition, and retrain the PINN deformation simulation model to reduce uncertainty error.

[0129] Based on the accuracy requirements of optical components, the allowable range of the confidence interval is set: the 95% confidence interval width of the PV value is ≤0.03μm, and the 95% confidence interval width of the RMS value is ≤0.01μm; if either indicator exceeds this range, the error compensation process is triggered.

[0130] Physical loss weight adjustment: For load conditions that exceed the range (such as impact loads and high-frequency periodic loads), the physical loss weight of the physical dominant network is increased by 5%-10% on the original basis (e.g., the original weight of 70% is adjusted to 75%-80%) to strengthen the role of physical constraints in suppressing uncertainty.

[0131] Add some data:

[0132] Supplementary areas: Focus on areas significantly affected by time-varying loads (such as high-stress areas corresponding to the peak moment of impact loads, areas of concentrated thermal strain within temperature fluctuation cycles) and areas with residuals > 1e-5.

[0133] Supplementation ratio: Increase the number of supplementary spots by 20%-30% based on the original number of spots. The density of the supplementary spots is consistent with the dynamic adaptive spotting rule in S400 (the density of the effective optical area is twice that of the non-optical area).

[0134] Retraining process: The dual-network alternating training logic of S222 is followed. The physical dominant network adopts the adjusted weights and the supplemented data, and the number of training rounds is 50% of the original number of training rounds (e.g., if the original was 1000 generations, retrain for 500 generations). The data correction network maintains the original configuration and only adapts to the new initial displacement field. The termination condition for retraining is: the confidence interval width ≤ the allowable range, and the physical loss ≤ 1e-5 and the data loss ≤ 1e-6.

[0135] This step, through a closed-loop mechanism of "confidence interval monitoring - weight adjustment - data point supplementation - retraining," achieves dynamic compensation for uncertainty, solving the problem that traditional methods cannot specifically reduce errors. The directional adjustment of physical loss weights and the precise supplementation of data points enhance physical rationality while avoiding computational waste caused by global retraining. Clear allowable ranges and compensation rules ensure the reproducibility of the operation, ultimately reducing the uncertainty error of the PINN deformation simulation model under time-varying load conditions by more than 30%. This ensures both global physical rationality and meets the high-precision requirements of the effective optical area, providing more reliable simulation support for the design of optical components under dynamic conditions.

[0136] S300, based on the clearly defined boundaries of the 3D model library, defines complex support conditions and load conditions as boundary conditions, constructs the PINN total loss function which includes physical loss, boundary loss and data loss, and uses the intra-domain and boundary point-scattering methods to generate spatial coordinate point-scattering data for training the PINN deformation simulation model, and constructs a 3D simulation database.

[0137] Boundary based on S100 3D model library Two types of boundary conditions are defined: the contact area between the support fixture and the optical element is a Dirichlet boundary condition (where displacement u=0, i.e., no displacement at the support); the surface subjected to acceleration or thermal loads is a Neumann boundary condition (where stress σ is specified). n =ρa or heat flux density Example: Thermal load acting surface q=500W / m²).

[0138] Additional constraints are applied to the lens frame contact area of ​​the optical element to limit radial displacement (u). x =0、u y =0), to ensure consistency with actual assembly conditions.

[0139] Furthermore, step S300, which involves generating spatial coordinate point data for training the PINN deformation simulation model using intra-domain and boundary point-scattering methods to construct a three-dimensional simulation database, includes the following steps:

[0140] S310: Based on the 3D model library, the effective optical region and the non-optical region are automatically divided. The initial point density of the effective optical region is set to the first preset value β1 times that of the non-optical region. Forced point density areas are preset at the edges of the effective optical region and at the curvature change points. The point density of the forced point density areas is the second preset value β2 times that of the initial point density of the effective optical region; 1 < β2 < β1.

[0141] The system automatically defines the regions of solid structures based on a 3D model library: the effective optical region of a reflector is the core area of ​​the mirror surface, with a radius equal to 90% of the total radius of the component (e.g., for a reflector with a total radius of 0.5m, the effective region radius is 0.45m); the effective optical region of a lens is the light-transmitting area, excluding the portion obstructed by the lens frame (e.g., for a lens with a diameter of 100mm, if the lens frame obstruction width is 5mm, the effective region diameter is 90mm); non-optical regions include non-core working areas such as component edges, lens frame connections, and support contact areas. The first preset value β1 = 2, and the second preset value β2 = 1.5 (satisfying 1 < β2 < β1).

[0142] The base dot density in non-optical areas is fixed at 50 dots / mm. 3 Initial dot density of the effective optical area = 50 × β1 = 100 dots / mm 3The dot density in the forced dot area (within 5mm of the edge of the effective optical area, at points of abrupt curvature change, such as the transition zone between the spherical and planar surfaces of a lens) is 100 × β² = 150 dots / mm. 3 A uniform random sampling method is used to generate spatial coordinate points in each region at a set density, ensuring uniform coverage without obvious clusters or gaps. In the forced sampling area, a grid-like sampling method is used, with the spacing between adjacent points equal to 1 / (sampling density). 1 / 2 (e.g., 150 points / mm) 3 (Corresponding spacing ≈ 0.081mm) to ensure accurate transmission of boundary constraints.

[0143] This step sets differentiated dot density based on the functional partitioning of optical components. High-density dotting in the effective optical area ensures the simulation accuracy of the core working area, while low-density dotting in the non-optical area controls the computational cost, solving the problem of "insufficient accuracy in key areas or waste across the entire area" in traditional uniform dotting. Forced high-density grid dotting in the dotting area ensures the accuracy of constraint transfer at boundary conditions and curvature abrupt changes, avoiding the failure of physical constraints at these key locations due to insufficient dotting. Overall, it achieves an initial dotting configuration that prioritizes accuracy while taking efficiency into account, laying a reasonable foundation for subsequent dynamic adjustments.

[0144] S311, after completing the training of the first preset number N1 generations of PINN deformation simulation model, calculate the physical residual and data residual of each point; the physical residual is the residual of the elasticity equilibrium differential equation and the residual of the heat conduction differential equation, and the data residual is the deviation between the model output and the high-precision data at the point. For the region where the residual is greater than the first preset residual threshold η1, the point density is increased, and the point density of the increased region is β1 times the original density.

[0145] The first preset number of generations is N1=50, the first preset residual threshold is η1=1e-5 (unit: μm), and the encryption factor is β1=2.

[0146] Residual calculation method: Physical residual = Mean square value of the residual of the elasticity equilibrium differential equation + Mean square value of the residual of the heat conduction differential equation (i.e. ); Data residual = Euclidean distance between the three-dimensional displacement field output by the model at the sampling point and the high-precision finite element / experimental data ([(u x _pred-u x _ref)²+(u y _pred-u y _ref) 2 +(u_z_pred-u_z_ref) 2 ] 1 / 2 The final residual is determined as max(physical residual, data residual), and the larger of the two is taken as the basis for judging the regional residual.

[0147] After every 50 generations of model training, all data points are traversed to calculate and determine the residuals. Continuous regions with residuals greater than 1e-5 are identified using spatial interpolation methods (such as Kriging interpolation). Data points in these regions are then densified, with the density of the densified data points equal to twice the density of the original region. The densified data points do not overlap with the original data points and uniformly fill the gaps in the region.

[0148] This step uses a mechanism of "training feedback every 50 generations + residual judgment" to accurately locate areas with insufficient simulation accuracy (high residual areas) and perform targeted point densification, avoiding the problems of "blind densification or untimely densification" in traditional point densification. The residual calculation takes into account both physical rationality and data fitting accuracy, ensuring that the densified areas are the key parts that truly need to improve simulation quality.

[0149] S312, for regions where the residual of the second preset number N2 generations is less than the second preset residual threshold η2, the points of the first preset ratio γ1 are clipped, and all points of the forced point area are retained during the clipping process; 0 < γ1 < 1; η1 > η2.

[0150] The second preset number of generations N2 = 100, the second preset residual threshold η2 = 1e-6 (unit: μm), the first preset ratio γ1 = 50% (satisfying 0 < γ1 < 1), and η1 = 1e-5 > η2 = 1e-6. Traverse all the scattered points and record the judgment residual for 100 consecutive generations (calculation method is the same as S311). If the judgment residual of all scattered points in a certain region is less than 1e-6 for 100 consecutive generations, it is judged as a redundant region. The redundant region does not include the forced scattered point region (the forced scattered point region is not judged as redundant regardless of the residual size).

[0151] For redundant areas, the number of points is reduced by 50% using a "uniform elimination" strategy—the points within the area are evenly divided into several groups according to spatial coordinates, with each group having an equal number of points, and one group is eliminated; during the reduction process, all points in the forced point area are strictly retained to ensure that the point density at boundary constraints and curvature abrupt changes is not reduced.

[0152] This step identifies redundant regions through "continuous residual monitoring for 100 generations," ensuring that the trimmed regions are those where the simulation accuracy has reached the standard and is stable, thus avoiding accuracy regression caused by premature trimming. The 50% uniform trimming ratio significantly reduces the number of invalid points while ensuring that the simulation accuracy does not decrease, thereby reducing the computational load and storage costs of model training. The forced retention rules for the trimmed regions ensure that the simulation reliability at boundary conditions and key structures is not affected by trimming, achieving a balance between "reduced computational resources and stable high accuracy."

[0153] S313 summarizes the dynamically adjusted data points and constructs a three-dimensional simulation database covering the entire physical computing domain.

[0154] Collect all data from the initial S310 point scattering, the S311 refined point scattering, and the remaining points after S312 trimming. The data includes: point spatial coordinates (x, y, z), region identifiers (effective optical region / non-optical region / forced point scattering region), corresponding boundary condition labels (Dirichlet / Neumann boundary), and high-precision reference data (finite element / experimental displacement field).

[0155] The data points are deduplicated (points with duplicate coordinates are removed) and outliers are removed (points with a residual greater than 1e-4 are removed). The data is divided into a training set (for model training) and a validation set (for model evaluation) in a 7:3 ratio, maintaining the same proportion of points in each region as the overall set (e.g., 80% of points in the effective optical region in both the training and validation sets). The 3D simulation database is stored in HDF5 format, organized hierarchically as "training set / validation set → region identifier → data type" (e.g., "3D simulation database / training set / effective optical region / coordinate data.h5"). The database includes a data index table that records the unique ID, coordinates, and associated parameters of each point, supporting fast retrieval by region and residual range.

[0156] This step summarizes the dynamically adjusted data points, ensuring that the database covers the entire physical computing domain. It includes both effective optical region data for high-precision requirements and necessary data for non-core regions, guaranteeing the comprehensiveness of model training. Data preprocessing and hierarchical storage improve the efficiency of data reading and use, avoiding data redundancy or missing data during training. The 7:3 training set to validation set division conforms to the conventional standards of machine learning, ensuring that the model can effectively fit the data and has good generalization ability. The final 3D simulation database provides high-quality data support for the efficient and high-precision training of the PINN deformation simulation model.

[0157] Furthermore, the S300 constructs a PINN total loss function comprising three parts: physical loss, boundary loss, and data loss, including the following steps:

[0158] S320, obtain the PINN total loss function L_Total=λ_Physics×L_Physics+λ_Conds×L_Conds+λ_Data×L_Data; where λ_Physics, λ_Conds, and λ_Data are the weight coefficients of physical loss L_Physics, boundary loss L_Conds, and data loss L_Data, respectively.

[0159] In this embodiment, the physical loss weight λ_Physics = 0.7 (core constraint, ensuring physical rationality), the boundary loss weight λ_Conds = 0.2 (adapting to actual working conditions), and the data loss weight λ_Data = 0.1 (aiding in improving accuracy). L_Physics is the sum of the residuals of the elasticity equilibrium differential equation and the heat conduction differential equation; L_Data is the sum of the residuals of the deviation between the model output and the finite element / experimental high-precision data within the effective optical region; and L_Conds is the sum of the residuals of the degree to which the boundary conditions are satisfied.

[0160] S321 uses a Gradient Balancing strategy to dynamically adjust the weight coefficients. During training, the convergence speed of each loss term is monitored in real time. If the convergence speed of a certain loss term is less than 1 / 3 of that of other loss terms, its corresponding weight coefficient is increased by 20%-30%.

[0161] Convergence speed is quantified by the "loss reduction rate per 50 generations," calculated as: Reduction rate = (Initial loss of the first 50 generations - Loss at the end of the current 50 generations) / Initial loss of the first 50 generations. During training, the reduction rates of the three loss terms are calculated in real-time. If the reduction rate of one loss term is less than 1 / 3 of the lowest value of the other two (e.g., L_Physics reduction rate = 0.1, L_Conds and L_Data reduction rates are both ≥ 0.3), its weight coefficient is increased by 20%-30% (e.g., λ_Physics increases from 0.7 to 0.84, i.e., 0.7 × 1.2). After adjustment, the sum of the three weight coefficients is kept at 1 (e.g., when increasing λ_Physics, λ_Conds and λ_Data are reduced proportionally, λ_Conds from 0.2 to 0.16, and λ_Data from 0.1 to 0.08). This process is monitored and dynamically adjusted every 50 generations until training ends.

[0162] S322, set the contact area between the support fixture and the optical element as a Dirichlet boundary condition, set the acceleration load or thermal load as a Neumann boundary condition, set constraints in the lens frame contact area of ​​the optical element, and incorporate the above boundary conditions into the calculation of boundary loss L_Conds.

[0163] Boundary conditions are strictly adapted to actual working conditions: the contact area between the support fixture and the optical element is set as a Dirichlet boundary condition, specifying that the displacement in this area is 0 (no relative movement); the surface acting on the acceleration load is set as a Neumann boundary condition, specifying that the stress value on this surface = ρa (ρ is the element density, a is the acceleration load); the surface acting on the thermal load is set as a Neumann boundary condition, specifying that the heat flux density on this surface = k × thermal load gradient (k is the element thermal conductivity); a radial displacement constraint is added to the lens frame contact area of ​​the optical element, specifying that the displacement in the x and y directions in this area is 0 (only axial deformation is allowed). The calculation logic of the boundary loss L_Conds is as follows: sum the residuals of "model output value - theoretical boundary condition value" at all boundaries, and take the mean square sum as the final L_Conds.

[0164] S400 calls upon the deformation control equation model library, PINN total loss function, and 3D simulation database, and uses physical information neural network for multi-condition generalization training to establish a correlation mapping database between various parameters in the parameter library and the surface accuracy index of optical components.

[0165] The PINN model corresponding to the target support structure is called from the deformation control equation model library, and the training set of the PINN total loss function and the 3D simulation database is loaded. The Adam optimizer is used, with the learning rate set to 1e-4, the batch size to 2048, and the number of training epochs to 1000. Training is stopped when either the total loss function value is ≤1e-6 or the loss decrease is ≤1e-8 over 200 consecutive epochs.

[0166] During the training process, the correspondence between each set of input parameters (design parameters + support structure parameters + load parameters) and the output surface accuracy index (PV value, RMS value) is recorded. Example: "R=0.5m, T=10mm, 3-point support, a=5g→PV=0.12μm, RMS=0.03μm";

[0167] The above mapping relationship is stored in the form of key-value pairs of "parameter combination - accuracy index", managed by SQLite database, and supports index query by parameter type (such as "number of support points" and "thermal load gradient") to form a machine learning parameter library.

[0168] This step, through explicit training parameter configuration, achieves standardized and reproducible model training, avoiding the instability of results caused by the ambiguity of the training process in traditional machine learning. The establishment of the mapping relationship directly links the input parameters and core performance indicators, solving the inefficiency problem of needing to recalculate to obtain results for different parameter combinations in traditional simulations. The structured storage and indexing function of the machine learning parameter library supports quick querying of the surface accuracy corresponding to specific parameter combinations, providing efficient data retrieval capabilities for subsequent model selection and design optimization, and significantly shortening the parameter evaluation cycle.

[0169] Furthermore, step S400 also constructs a bidirectional PINN model of forward prediction and backward inversion, specifically implemented through the following steps:

[0170] S410, forward model training, takes design parameters, support structure parameters, and external load parameters from the parameter library as input, and outputs surface deformation data of optical elements; the surface deformation data includes PV value, RMS value and three-dimensional displacement field, and establishes a forward mapping relationship.

[0171] Select complete combinations of design parameters (radius R, thickness T, radius of curvature C), support structure parameters (support point location, tooling material parameters), and external load parameters (thermal load gradient, acceleration load, including time-varying load) from the parameter library. Input parameters need to be standardized (x'=(x-μ) / σ, where μ is the mean of the parameter library and σ is the standard deviation). The forward model is the PINN deformation simulation model (dual-network collaborative architecture + time-varying load adaptive module) constructed by S200, which adopts a two-stage optimization algorithm (Adam+L-BFGS), with a learning rate of 1e-4, a batch size of 2048, and 1500 training epochs; the convergence condition is that the total loss is ≤1e-6, and the error between the training set and the validation set for PV and RMS values ​​is ≤3%.

[0172] The model directly outputs a three-dimensional displacement field (u x u y PV value is calculated as "maximum value - minimum value of displacement field within effective optical region" (e.g., displacement field range 0.02~0.11μm, PV=0.09μm); RMS value is calculated as "root mean square of displacement field within effective optical region" (RMS=[(1 / N)×Σu_z)). 2 ] 1 / 2 (N is the number of effective area points).

[0173] The input parameter combinations and corresponding output surface deformation data are stored in the form of "key-value pairs" to form a forward mapping library, which supports quick query of deformation results by parameters.

[0174] The forward model in this step is seamlessly integrated with the previous PINN deformation simulation model, ensuring the consistency and reproducibility of the model. The standardized training conditions and output data calculation rules make the forward mapping relationship accurate and reliable, providing a high-quality basic framework for the inverse model. The surface deformation data covers the displacement field and core accuracy indicators (PV / RMS), fully meeting the input requirements of inverse inversion, solving the limitation of traditional forward simulation that only outputs a single result and cannot support inversion, while improving the training efficiency by more than 10 times compared with the traditional finite element method.

[0175] S411, Construct a reverse model based on the forward model, fix the known parameters, and input the measured surface error data; the measured surface error data includes PV value, RMS value and Zernike coefficient, and output unknown parameters, including unknown thermal load gradient, support point position deviation and tooling stiffness.

[0176] The inverse model fully adopts the network architecture of the forward model (dual network collaboration + time-varying load module), only reversing the input and output dimensions. The network size, activation function (Tanh), and initialization method (He initialization) are consistent with the forward model, ensuring model compatibility. Design parameters (such as R=0.5m, T=10mm) and optical element material property parameters (such as E=72GPa for fused silica) in the parameter library are fixed, and only unknown parameters are retained as the output to be solved by the model.

[0177] The input is the measured surface error data—obtained by measuring the actual component using a Zygo GPI XPS interferometer, with PV and RMS values ​​having an accuracy of ≤0.01μm. The Zernike coefficients are taken from the first 37th order (to comprehensively describe the surface deviation). The output is unknown parameters—unknown thermal load gradient (range -50~50℃ / m), support point position deviation (range ±10mm), and tooling stiffness (range 100~300GPa). The output results must be within the physically reasonable range.

[0178] The Adam optimizer was used with a learning rate of 5e-5, a batch size of 1024, and 800 training epochs. The initial loss function was mainly based on data loss (the deviation between the model output and the true unknown parameters), with a weighting of 70%.

[0179] This step builds an inverse model based on the forward model, avoiding the repetitive work of redesigning the network and greatly improving modeling efficiency. The input measured surface error data includes Zernike coefficients, which are more comprehensive than a single PV / RMS value, improving the accuracy of the inversion. The output unknown parameters directly address engineering pain points (such as unknown loads and support deviations), solving the core bottleneck that traditional finite element methods cannot easily invert such parameters, and providing a direct means for fault diagnosis and parameter optimization of optical components.

[0180] S412 employs a two-way constraint mechanism, using the output of the forward model as virtual measured data for pre-training of the inverse model. The inversion results of the inverse model are fed back to the forward model for verification. If the relative deviation between the surface error corresponding to the inversion result and the measured surface error is less than or equal to a preset deviation threshold, the inversion is deemed valid.

[0181] Inverse model pre-training: 100 sets of input parameters are randomly selected from the forward mapping library. The corresponding surface deformation data (PV, RMS, 3D displacement field) are output by the forward model and converted into "virtual measured data" (supplemented with Zernike coefficients, obtained by fitting the displacement field). This virtual data is then input into the inverse model and pre-trained for 500 generations to enable the inverse model to initially grasp the mapping logic of "surface data → parameters". The pre-training convergence condition is that the virtual data inversion error is ≤5%.

[0182] Inversion result feedback verification: Input the measured surface error data into the pre-trained inverse model to obtain the inversion unknown parameters; input the inversion parameters into the forward model to generate the corresponding surface deformation prediction data; calculate the relative deviation between the prediction data and the measured surface error data (deviation = |predicted value - measured value| / measured value × 100%).

[0183] Preset deviation threshold and judgment rules: The preset deviation threshold is 5%. If the relative deviation of PV value and RMS value is ≤5% and the average relative deviation of Zernike coefficient is ≤8%, the inversion is deemed valid. If the threshold is exceeded, the deviation is fed back to the inverse model, the learning rate is adjusted (increased by 20%) and retrained for 300 generations, and the verification is repeated until the conditions are met.

[0184] The bidirectional constraint mechanism in this step forms a closed loop of "forward pre-training → backward inversion → forward verification", which effectively reduces the training difficulty of the inverse model and improves the reliability of the inversion results. The clear deviation threshold and verification rules avoid subjective judgment of the inversion results and ensure the effectiveness of the inversion. The feedback adjustment mechanism enables the model to adaptively correct errors, which solves the problems of "unstable training and large inversion deviation" of traditional inverse models and greatly improves the accuracy and efficiency of solving inverse problems.

[0185] S413 introduces physical regularization and sparse regularization terms into the loss function of the inverse model. The physical regularization term constrains the inversion parameters to satisfy the mechanical equilibrium condition, and the sparse regularization term constrains the spatiotemporal sparsity of the unknown load, thus avoiding the multiple solutions of the inverse problem.

[0186] In this embodiment, the physical regularization term L_PhysReg = mean (residual of the mechanical equilibrium equation) 2 Substitute the unknown parameters obtained from the inversion (such as the deviation of the support point position and the stiffness of the tooling) into the elasticity equilibrium differential equation. The residuals of the calculation equations are constrained to ensure that the inversion parameters meet mechanical equilibrium, so as to avoid solutions that are "parameterically feasible but physically contradictory"; the physical regularization term accounts for 30% of the weight in the loss function of the inverse model.

[0187] The sparse regularization term L_SparseReg=λ×||w||1 (λ=0.01), where w is the time series vector of the unknown load (such as the unknown thermal load gradient). The L1 regularization constrains the spatiotemporal sparsity of the unknown load (such as the impact load having a non-zero value only at the peak time and 0 at other times), reducing redundant solutions. The sparse regularization term accounts for 20% of the weight in the loss function.

[0188] L_Reverse = 0.5 × L_Data + 0.3 × L_PhysReg + 0.2 × L_SparseReg, where L_Data is the sum of the squared deviations of the inversion parameters from the true values ​​(or verification values), ensuring a balance between accuracy and constraints.

[0189] The physical regularization term introduced in this step fundamentally constrains the physical rationality of the inversion parameters, avoiding the multiple solutions situation in inverse problems that are "mathematically feasible but ineffective in engineering". The sparse regularization term specifically constrains the spatiotemporal distribution characteristics of the unknown load, further compressing the solution space and ensuring the uniqueness of the inversion result. The weight allocation of the regularization term takes into account both accuracy and constraint, enabling the inverse model to solve the core pain points of "multiple solutions and slow convergence" in traditional inverse problems while performing efficient inversion, significantly improving the engineering practical value of the inversion result.

[0190] S500: Based on the aforementioned association mapping database, and combined with the deformation control equation model library and the three-dimensional simulation database, the PINN deformation simulation model is refined and its physical constraints are verified. Target PINN deformation simulation models that meet the preset conditions are then selected.

[0191] Deformation control equation model library call: Based on the type of optical element (such as 0.5m diameter fused silica mirror) and the type of support structure (3-point support + aluminum alloy tooling), the corresponding PINN deformation control equation model is retrieved from the model library as the basic physical framework for training, ensuring that the model is initially adapted to the mechanical / thermal constraint logic of the element.

[0192] 3D simulation database access: Data is divided according to the ratio of "training set: validation set: test set = 7:2:1". The training set selects 100 points / mm of effective optical area (radius 0.45m). 3 Scattered data + 50 points / mm in non-optical areas 3 The dataset is divided into several sets. The validation set is used to monitor overfitting in real time, and the test set contains 10 typical working conditions that were not involved in the training (such as 5g acceleration + 30℃ / m thermal load). All data reuses the preprocessing results of the database that have been deduplicated and removed outliers.

[0193] Machine learning parameter library call: retrieve standardized hyperparameters (Adam+L-BFGS optimizer, initial learning rate 1e-4, batch size 2048), initial weights of the loss function (λ_Physics=0.7, λ_Conds=0.2, λ_Data=0.1), and dynamic adjustment rules (such as convergence speed threshold for Gradient Balancing, and point scattering / pruning parameters) to unify training configuration.

[0194] The multi-database collaborative training process is as follows:

[0195] Initialization training: Input the training set data from the 3D simulation database into the retrieved deformation control equation model, start training according to the machine learning parameter library configuration, use the Adam optimizer for fast iteration for the first 500 generations, and switch to the L-BFGS optimizer for fine convergence for the next 1000 generations; calculate the physical residual (elasticity / heat conduction equation residual) every 50 generations during training to ensure that the model output meets the physical constraints of the deformation control equation.

[0196] Dynamic adaptation and adjustment: Real-time adjustment according to machine learning parameter library rules: If the convergence speed of physical loss is only 1 / 4 (<1 / 3) of that of data loss, then increase λ_Physics from 0.7 to 0.84; If the residual of a certain area is 1.2e-5 (>η1=1e-5), retrieve the denser points for that area from the 3D simulation database (density increased to 200 points / mm). 3 Supplementary training; if the residual of a certain region is 8e-7 for 100 consecutive generations (<η2=1e-6), cut off 50% of the redundant scattered points;

[0197] Intermediate model saving: The intermediate model is saved every 200 generations, recording key indicators such as loss value, PV / RMS prediction error, etc., to avoid training interruption and loss of progress.

[0198] The preset filtering conditions and execution are as follows:

[0199] Preset conditions are defined as follows: core accuracy conditions (relative deviation between effective optical area PV / RMS value and measured data ≤3%, displacement field average absolute error ≤0.01μm), physical constraints (physical loss ≤1e-5, global physical equation residual ≤1e-4), and efficiency stability conditions (simulation time for single working condition ≤10s, test set loss fluctuation ≤5%).

[0200] Screening process: First, intermediate models that do not meet the accuracy / physical constraints are eliminated (e.g., a model with a PV deviation of 4.5% is directly eliminated). Then, the efficiency and stability of the candidate models are verified and sorted according to "accuracy first, efficiency second". Finally, the optimal model that meets the requirements of all working conditions in the test set is selected as the target PINN deformation simulation model.

[0201] Archive closed loop: Archive the parameters and performance indicators of the target model into the deformation control equation model library to form a reusable standardized model.

[0202] This step addresses the core issues of traditional PINN training—namely, "disorganized data, arbitrary parameters, and lack of physical constraints"—through deep collaboration among deformation control equation model libraries, 3D simulation databases, and machine learning parameter libraries. Multi-library fusion ensures that training is supported by both a precise physical framework and high-quality data and standardized parameters, significantly improving the efficiency and reproducibility of model training. Layered selection criteria balance accuracy, physical rationality, efficiency, and stability, avoiding the problems of "pursuing accuracy at the expense of physical laws" or "meeting physical constraints but lacking accuracy." The final selected target model conforms to objective physical laws and meets the high-precision, high-efficiency requirements of engineering scenarios, achieving more than 10 times the speedup of traditional finite element simulation with errors controlled within 3%, providing a standardized and highly reliable simulation tool for the design, optimization, and fault diagnosis of optical components.

[0203] S600 inputs different combinations of design parameters, support structure parameters, and given external load parameters into the target PINN model to obtain the key performance prediction results of the optical element.

[0204] Design parameters (radius R: 0.5m~2m, thickness T: 5mm~20mm, radius of curvature C: 5m~∞), support structure parameters (number of support points: 3 / 9 / 12 points, tooling material: aluminum alloy / titanium alloy / stainless steel, support point offset: 0mm±10mm), and external load parameters (thermal load gradient: -50℃ / m~50℃ / m, acceleration load: 0g~10g) are selected from the parameter library. 300~500 sets of non-repeating parameter combinations are generated using Latin hypercube sampling, covering typical working conditions such as "small diameter + weak load", "large diameter + strong load", and "special support + time-varying load" (example combination: R=0.8m, T=12mm, 9-point support, titanium alloy tooling, thermal load gradient 25℃ / m, acceleration load 6g).

[0205] The standardization rule of S100 is followed (x'=(x-μ) / σ), where μ and σ are the mean and standard deviation of the corresponding parameters in the parameter library (e.g., μ=1.25m and σ=0.43m for R), ensuring that the input parameters are of the same magnitude and match the input format during target model training. Physically infeasible parameter combinations are eliminated (e.g., large-aperture thin mirrors with T=5mm and R=2m, exceeding the material stiffness limit), retaining approximately 300 effective combinations.

[0206] The target PINN model (.pth format) archived by S500 is invoked, and the "inference mode" of the deep learning framework is started (gradient calculation and batch normalization updates are disabled, and only forward propagation is retained) to ensure inference efficiency. The standardized parameter combinations are grouped into batches of 128 and input into the model. The model outputs the three-dimensional displacement field (u) of the effective optical region. x u y (e.g., u_z); single-group parameter inference time ≤ 10s, total time for 300 combinations ≤ 50min (can be achieved with ordinary server configuration).

[0207] Based on the displacement field, the core index is derived as follows: PV value = maximum effective displacement - minimum effective displacement (e.g., if u_z ranges from 0.03 to 0.15 μm, PV = 0.12 μm); RMS value = [(1 / N) × Σu_z] 2 ] 1 / 2 (N is the number of points to be sprinkled in the effective area, for example N=5×10) 6 RMS = 0.04 μm); 37th order Zernike coefficients (obtained by fitting the displacement field with Zernike polynomials, characterizing the type of surface distortion, such as coma and spherical aberration).

[0208] With "parameter combination - performance index" as the core, generate an Excel format report, which includes detailed values ​​of each parameter group, PV value, RMS value, first 10 Zernike coefficients, and whether the design requirements are met (e.g., preset PV≤0.2μm is qualified).

[0209] Visualization: Displacement field cloud plots and trend curves of PV / RMS values ​​as a function of parameters (such as the growth curve of RMS value as a function of acceleration load) are generated using Matplotlib or Paraview, which intuitively show the influence of parameters on performance.

[0210] Result verification: 10 sets of parameters were randomly selected to correspond to actual processed components. The performance was measured by Zygo GPI XPS interferometer. If the relative deviation between the predicted value and the measured value was ≤3%, the prediction result was deemed valid.

[0211] This step supports rapid inference using multi-dimensional parameter combinations, addressing the core pain points of traditional finite element methods: "time-consuming single-condition modeling and solving (several hours / day) and difficulty in covering multiple design schemes." Standardized parameter processing and inference workflows ensure the reproducibility of prediction results, with a deviation accuracy within 3% meeting engineering design requirements. The output performance indicators include both core accuracy parameters (PV / RMS) and surface distortion characteristics (Zernike coefficients), comprehensively supporting optical component design iteration, support scheme optimization, and load compensation decisions. Visualized reports and trend analysis help designers quickly locate key influencing parameters (such as the sensitivity of the number of support points to the PV value), significantly shortening the R&D cycle and reducing trial-and-error costs. Compared to traditional R&D processes, efficiency is improved by 5-10 times, providing efficient and reliable technical support for the rapid deployment of high-precision optical systems.

[0212] Furthermore, although the steps of the method in this disclosure are described in a specific order in the accompanying drawings, this does not require or imply that the steps must be performed in that specific order, or that all the steps shown must be performed to achieve the desired result. Additional or alternative steps may be omitted, multiple steps may be combined into one step, and / or a step may be broken down into multiple steps.

[0213] While specific embodiments of the invention have been described in detail by way of examples, those skilled in the art should understand that the examples are for illustrative purposes only and are not intended to limit the scope of the invention. Those skilled in the art should also understand that various modifications can be made to the embodiments without departing from the scope and spirit of the invention.

Claims

1. A method for simulating the surface profile of optical components based on a physical information neural network, characterized in that, The process includes the following steps: S100, constructing a feedforward neural network using a physical information neural network method based on a 3D model library corresponding to the optical element; the input of the feedforward neural network is spatial coordinates, time and load parameters, and the output is a 3D displacement field; the 3D model library is obtained from a parameter library corresponding to the optical element; the parameter library includes the design parameters of the optical element, support structure parameters, material property parameters, and external load parameters; S200, embedding the physical field differential equation describing the deformation of the optical element into the loss function of the neural network as a physical constraint, constructing a corresponding PINN deformation simulation model according to different support structure parameters, and generating a deformation control equation model library; S300, based on the three... The model library has clearly defined boundaries, and complex support and load conditions are defined as boundary conditions. A PINN total loss function, comprising physical loss, boundary loss, and data loss, is constructed. Spatial coordinate point data for training the PINN deformation simulation model is generated using intra-domain and boundary point-scattering methods, thus constructing a 3D simulation database. In step S400, the deformation control equation model library, the PINN total loss function, and the 3D simulation database are called. Multi-condition generalization training is performed using a physical information neural network to establish a correlation mapping database between various parameters in the parameter library and the surface accuracy indicators of optical components. In step S500, based on the aforementioned correlation mapping database, and combined with the deformation control equation model library… Using a 3D simulation database, the PINN deformation simulation model is refined and its physical constraints are verified. Target PINN deformation simulation models meeting preset conditions are selected. In step S600, different combinations of design parameters, support structure parameters, and given external load parameters are input into the target PINN model to obtain the key performance prediction results of the optical element. The PINN deformation simulation model described in step S200 is a dual-network collaborative architecture of a physical dominant network and a data correction network. Specifically, it includes the following steps: In step S220, a physical dominant network is constructed by inputting spatial coordinates, time, load parameters, and optical element material parameters, outputting a preliminary 3D displacement field, and setting the loss function of the physical dominant network. The function is primarily based on physical loss L_Physics, with an initial weighting of physical loss ≥ α1. The physical loss is calculated from the residuals of the elasticity equilibrium differential equation and the heat conduction differential equation; α1 is the first preset weighting threshold. S221, a data correction network is constructed, using the preliminary three-dimensional displacement field output by the physical-dominant network as input, combined with finite element high-precision simulation data or experimental measured data of the effective optical region of the optical element, to output the final three-dimensional displacement field. The loss function of the data correction network is set primarily based on data loss L_Data, with an initial weighting of data loss ≥ α2. Correction loss is calculated only within the effective optical region; α2 is the second preset weighting threshold; α1 < α2.S222 involves alternating training of the physical dominant network and the data correction network. The training results of the physical dominant network are used as the initial input to the data correction network. The correction error of the data correction network is fed back to the physical dominant network through a backpropagation mechanism, dynamically adjusting the physical loss weights of the physical dominant network to achieve a balance between global physical plausibility and local high precision in the effective optical region.

2. The method according to claim 1, characterized in that, Step S200 The process includes the following steps: S210, setting and initializing the structure of the feedforward neural network; wherein the feedforward neural network has 6-12 layers, each layer has 60-100 neurons, Tanh or Sigmoid is selected as the activation function, the network weights and biases are initialized using the He initialization method, and the network forward propagation process is constructed; S211: introducing the elasticity equilibrium differential equation and the heat conduction differential equation into the physical loss function; wherein the stress tensor in the elasticity equilibrium differential equation is related to the three-dimensional displacement field through the thermoelastic constitutive relation; S212: obtaining the first and second order partial derivatives of the feedforward neural network output with respect to the input spatial coordinates, substituting them into the physical field differential equation to calculate the residuals, to ensure that the neural network output meets the physical constraints; S213: instantiating the corresponding differential equation models for the support structure parameters, including different support point positions and support tooling material parameters, to obtain the corresponding PINN deformation simulation models, thereby generating a deformation control equation model library.

3. The method according to claim 1, characterized in that, The load parameters mentioned in step S100 include time-varying loads. The PINN deformation simulation model in step S200 integrates a time-varying load adaptive modeling and uncertainty quantification module. Specifically, this includes the following steps: S230, adding a load time-varying feature extraction module to the input layer of the PINN deformation simulation model, using an attention mechanism to capture the time evolution features of thermal loads and acceleration loads; the time evolution features include the peak time of the impact load and the period of temperature fluctuations. Based on the feature extraction results, the time step size and activation function weights of the physical dominant network are dynamically adjusted to adapt to transient and periodic time-varying conditions; S2 31. A Bayesian physical information neural network architecture is adopted to quantify uncertainty. The weights and biases of the physical dominant network are defined as Gaussian probability distributions. Several sets of model parameter samples are generated through the Markov chain Monte Carlo sampling method. The 95% confidence interval of the surface accuracy index is output to quantify the impact of material parameter fluctuations and measurement errors on the simulation results. S232. It is determined whether the confidence interval exceeds the preset allowable range. If it does, the weight ratio of physical loss under the corresponding load condition is automatically increased, and the scattered data under the corresponding load condition is supplemented. The PINN deformation simulation model is retrained to reduce uncertainty error.

4. The method according to claim 2, characterized in that, Step S300 describes using intra-domain and boundary point-scattering methods to generate spatial coordinate point data for training the PINN deformation simulation model, and constructing a three-dimensional simulation database. The process includes the following steps: S310, automatically dividing the effective optical region and non-optical region based on the 3D model library, setting the initial point density of the effective optical region to be β1 times the first preset value of the non-optical region, and setting a forced point-spreading area at the edge of the effective optical region and at the curvature abrupt change, with the point density of the forced point-spreading area being β2 times the second preset value of the initial point density of the effective optical region; 1 < β2 < β1; S311, after completing the first preset number N1 generations of PINN deformation simulation model training, calculating the physical residual and data residual of each point; the physical residual is the elasticity equilibrium differential equation. The residuals are the residuals of the heat conduction differential equation. The data residuals are the deviations between the model output and the high-precision data at the point of application. Points are densified in the region where the residuals are greater than the first preset residual threshold η1. The density of points in the densified region is β1 times the original density. S312, for the region where the residuals are less than the second preset residual threshold η2 for a second preset number N2 generations, the points are clipped by the first preset ratio γ1. During the clipping process, all points in the forced point application area are retained. 0 < γ1 < 1; η1 > η2. S313, the dynamically adjusted point data are summarized to construct a three-dimensional simulation database covering the entire physical calculation domain.

5. The method according to claim 1, characterized in that, Step S400 also constructs a bidirectional PINN model for forward prediction and backward inversion, specifically including the following steps: S410, forward model training, inputting design parameters, support structure parameters, and external load parameters from the parameter library, and outputting surface deformation data of the optical element; the surface deformation data includes PV value, RMS value, and three-dimensional displacement field, establishing a forward mapping relationship; S411, constructing a backward model, based on the forward model, fixing known parameters, and inputting measured surface error data; the measured surface error data includes PV value, RMS value, and Zernike coefficient, outputting unknown parameters, the unknown... The parameters include the unknown thermal load gradient, support point position deviation, and tooling stiffness; S412, adopting a two-way constraint mechanism, the output of the forward model is used as virtual measured data for pre-training of the inverse model, and the inversion result of the inverse model is fed back to the forward model for verification. If the relative deviation between the surface error corresponding to the inversion result and the measured surface error is less than or equal to the preset deviation threshold, the inversion is deemed valid; S413, physical regularization term and sparse regularization term are introduced into the loss function of the inverse model. The physical regularization term constrains the inversion parameters to satisfy the mechanical equilibrium condition, and the sparse regularization term constrains the spatiotemporal sparsity of the unknown load to avoid the multiple solutions of the inverse problem.

6. The method according to claim 1, characterized in that, S300 constructs the PINN total loss function, which includes three parts: physical loss, boundary loss, and data loss. This involves the following steps: S320, obtaining the PINN total loss function L_Total = λ_Physics × L_Physics + λ_Conds × L_Conds + λ_Data × L_Data; where λ_Physics, λ_Conds, and λ_Data are the weight coefficients of the physical loss L_Physics, boundary loss L_Conds, and data loss L_Data, respectively; S321, dynamically adjusting the weight coefficients using a Gradient Balancing strategy, monitoring the convergence speed of each loss term in real time during training. If the convergence speed of a certain loss term is less than 1 / 3 of that of other loss terms, its corresponding weight coefficient is increased by 20%-30%; S322, setting the contact area between the support fixture and the optical element as a Dirichlet boundary condition, setting the acceleration load or thermal load as a Neumann boundary condition, setting constraints in the lens frame contact area of ​​the optical element, and incorporating the above boundary conditions into the calculation of the boundary loss L_Conds.

7. The method according to claim 2, characterized in that, In the elasticity equilibrium differential equation, the volume force f = ρa; where ρ is the density of the optical element and a is the acceleration load in the external load parameters; in the heat conduction differential equation, the internal heat source Q is defaulted to 0, and if the internal heat source is considered, Q is included in the external load parameter library.

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