A Machine Tool Thermal Field Reconstruction Method Based on Geometric Prior and Dual Adaptive PINN

Abstract

This invention discloses a machine tool thermal field reconstruction method based on geometric prior and dual adaptive PINN, comprising: collecting sparse temperature sensor data during machine tool operation; constructing a geometric perception module to extract surface point clouds and feature edges of machine tool components, calculate singularity distance fields, and perform region labeling; constructing a geometric prior adaptive sampling module to generate dynamically mixed sampling points based on distance fields and residual feedback; constructing a geometric perception adaptive weighting module to dynamically adjust the weights of each loss term using geometric difficulty factors and loss change rates; constructing a PINN model incorporating Gaussian Fourier eigenmaps, and training a joint loss function by integrating physical constraints, boundary conditions, and data constraints, ultimately achieving three-dimensional transient thermal field reconstruction of machine tool components. This invention achieves effective fusion of physical mechanisms and sparse data, thereby improving the accuracy of PINN in predicting thermal fields.

Description

Technical Field

[0001] This invention relates to the fields of precision manufacturing and industrial digital twin technology, and in particular to a method for reconstructing the thermal field of machine tools based on geometric priors and dual adaptive PINN. Background Technology

[0002] In the precision machining process of high-end CNC machine tools, thermal error is the largest single source of error affecting machining accuracy, accounting for approximately 40% to 70% of the total geometric error. Internal heat sources of the machine tool (such as motor heating, bearing friction, and cutting heat) can cause non-uniform three-dimensional temperature fields in structural components, leading to complex thermal deformation. To achieve high-precision thermal error compensation, it is first necessary to accurately obtain the real-time three-dimensional temperature field distribution of key machine tool components (such as stepped shafts and ball screws).

[0003] Existing methods for acquiring thermal fields are mainly divided into two categories: contact measurement methods and numerical simulation methods. Contact measurement methods typically involve placing temperature sensors on the machine tool surface, but due to limitations in installation space and cost, the number of sensors is extremely sparse, and they cannot measure the temperature inside components (such as the shaft), making it difficult to reconstruct a complete three-dimensional thermal field. Numerical simulation methods (such as the finite element method, FEM) rely on accurate three-dimensional modeling and parameter setting, and are time-consuming.

[0004] In recent years, data-driven methods based on deep learning have been widely applied. For example, Chinese patent document CN114442558A discloses a method for selecting machine tool temperature-sensitive point combinations based on a self-attention mechanism, including: S1, collecting temperature data and spindle thermal deformation during CNC machine tool operation; S2, inputting the temperature data into a multilayer perceptron based on a self-attention mechanism to obtain a primary temperature-sensitive point combination; S3, obtaining a secondary temperature-sensitive point combination based on the primary temperature-sensitive point combination and the spindle thermal deformation; S4, processing the secondary temperature-sensitive point combination and the spindle thermal deformation through a multilayer perceptron based on a self-attention mechanism to obtain the machine tool temperature-sensitive point combination. Chinese patent document CN121257261A discloses a method for modeling assembly process errors considering heat, using a physical information neural network to construct a predictive model for thermal deformation of assembly mating surfaces during component assembly.

[0005] However, standard multilayer perceptron (MLP) networks suffer from "spectral bias," meaning the network tends to prioritize learning low-frequency smooth features, making it difficult to fit high-frequency thermal gradient changes at common geometrical abrupt changes in machine tool parts (such as steps and keyways). Furthermore, existing physical information neural networks (PINNs) often employ uniform sampling or gradient-based adaptive sampling when dealing with complex geometric boundaries. The former results in insufficient sampling in critical regions (hotspot areas), while the latter lacks prior guidance in the early training stages, leading to wasted computational resources in non-critical regions (cold-start blind zones). Moreover, the weights of the loss function under different boundary conditions are difficult to balance, causing model convergence difficulties.

[0006] Therefore, there is an urgent need for a method that can guide the training of a physical model using geometric prior knowledge and dual adaptive optimization of sampling and weighting, under conditions of extremely sparse sensor data, so as to achieve high-precision and high-efficiency three-dimensional thermal field reconstruction of complex machine tool parts. Summary of the Invention

[0007] To address the technical challenge of accurately reconstructing high-frequency thermal gradients of complex geometries (such as machine tool spindles and lead screws) under sparse observation conditions, this invention provides a machine tool thermal field reconstruction method based on geometric priors and dual adaptive PINN. By driving the dynamic evolution of sampling distribution and loss weights through pre-calculated geometric singularity distance field, the method effectively integrates physical mechanisms and sparse data, thereby improving the accuracy of PINN-predicted thermal fields.

[0008] A method for reconstructing the thermal field of a machine tool based on geometric prior and dual adaptive PINN includes the following steps:

[0009] (1) Arrange several temperature sensors on the thermally sensitive parts of the machine tool components and collect sparse temperature data during the operation of the machine tool;

[0010] (2) Construct a geometric perception module to build a uniformly dense candidate sampling point cloud from the three-dimensional model of the machine tool component; extract the surface point cloud set and feature edge set, calculate the singularity distance field and surface distance field of each candidate sampling point and perform region marking;

[0011] (3) Construct an adaptive sampling module with geometric priors to dynamically select training points from the candidate sampling point cloud;

[0012] (4) Construct a geometry-aware adaptive weight module to assign geometric difficulty factors to different regions based on region labels and dynamically adjust the weights of each loss term during training.

[0013] (5) Construct the PINN model with Gaussian Fourier feature mapping, use the temperature dataset in step (1) and the training points in step (3) to construct the joint loss function, and train the PINN model by combining the adaptive sampling module and the adaptive weight module.

[0014] (6) After training, the temperature value of the machine tool component at any time and any position is inferred using the PINN model to complete the three-dimensional global thermal field reconstruction.

[0015] In step (2), the singularity distance field is the Euclidean distance from the sampling point to the nearest feature edge, and the surface distance field is the shortest Euclidean distance from the sampling point to the surface of the machine tool component.

[0016] For any candidate point, the expressions for calculating its singularity distance field and surface distance field are as follows:

[0017] ;

[0018] ;

[0019] In the formula, To compute any candidate point within the computation domain, For surface point cloud collection A little bit above, For the set of characteristic edges A little bit above, For the Euclidean norm, The distance from the characterizing point to the surface of the component is called the face-changing distance field. The distance from the representation point to the geometric abrupt change is the singularity distance field.

[0020] In step (2), a singularity distance threshold is set. and surface distance threshold Each candidate point is marked as a different region, including complex region, surface region and interior region.

[0021] In step (3), training points are dynamically selected from the candidate sampling point cloud, specifically including:

[0022] During the warm start phase of training, sampling is performed solely using the geometric sampling probability distribution, while during the adaptive phase of training, a hybrid sampling method combining the geometric sampling probability distribution and the residual sampling probability distribution is used.

[0023] The geometric sampling probability distribution includes the boundary sampling probability distribution and the volume domain sampling probability distribution;

[0024] For sampling points in the boundary computational domain, a boundary sampling probability distribution is constructed, which is inversely proportional to the singularity distance field. For sampling points in the internal computational domain, a volume domain sampling probability distribution is constructed, which follows a double distance decay law and is inversely proportional to both the surface distance field and the singularity distance field.

[0025] During the adaptive training phase, points from the candidate sampling point cloud are periodically input into the current PINN model to calculate the absolute value of the residual of its partial differential equation, and the residual sampling probability distribution is constructed based on this value.

[0026] The calculation expressions for the boundary sampling probability distribution and the volume domain sampling probability distribution are as follows:

[0027] ;

[0028] ;

[0029] In the formula, To compute any candidate point within the computation domain, Let be the boundary sampling probability distribution function. Let be the volume domain sampling probability distribution function. and These are the singularity distance field and surface distance field of the candidate points calculated in the geometry sensing module, respectively. and The normalization constant is To prevent division by zero constants, , , The exponential hyperparameter for controlling the clustering of sampling points, where The degree to which boundary sampling points cluster towards geometric abrupt changes. , The degree of aggregation of volume sampling points toward the surface and geometric abrupt changes is controlled respectively.

[0030] The expression for calculating the mixed sampling probability distribution in the adaptive sampling phase is:

[0031] ;

[0032] In the formula, To compute any candidate point within the computation domain, The mixed sampling probability distribution function, Let be the geometric sampling probability distribution function, if For boundary sampling points, then ,like If the sampling point is a volume region, then it is , Let be the probability distribution function of the residuals at the sampling points. and The weighting coefficients for the geometric probability distribution and the residual probability distribution are respectively, satisfying... .

[0033] In step (4), during the training process, the weights of each loss term are dynamically adjusted based on the historical rate of change of each loss term combined with the geometric difficulty factor.

[0034] The formula for calculating adaptive weights is:

[0035] ;

[0036] In the formula, Let be the relative weight of the k-th loss term in the t-th training period. The total number of loss items. For smoothing coefficients, Let k be the historical relative rate of change of the k-th loss term. The geometric difficulty factor for the k-th loss term is determined based on the region in step (1), with a larger value assigned to complex regions. Sum the exponential values ​​of the weighted rates of change for all loss components.

[0037] In step (5), the calculation expression for the Gaussian Fourier feature map is:

[0038] ;

[0039] In the formula, The frequency matrix has the following elements. It follows an isotropic Gaussian distribution , Control the spectrum range that the network can sense. These are normalized spacetime coordinates.

[0040] In step (5), a joint loss function is constructed by integrating physical constraints, boundary condition constraints, and sparse temperature data constraints.

[0041] The joint loss function is calculated as follows:

[0042] ;

[0043] In the formula, For PDE sampling domain, and The first The set of sampling points and dynamic weights for each PDE loss term. For PDE residuals, For the boundary condition sampling domain, and The first The set of sampling points and dynamic weights for each boundary condition loss term. For boundary condition residuals, For the weights of the data loss items, For sparse sensor numbers, and The first Predicted and actual temperatures at each sensor location. This represents the mean square error.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] 1. This invention deeply integrates sparse temperature sensor data, the physical mechanism of heat conduction, and geometric prior information based on singularity distance field (SDF). This combination overcomes the limitation of traditional contact measurement in acquiring the internal temperature field of machine tool components (such as the shaft center), while also overcoming the spectral bias problem of standard neural networks when processing complex geometric features such as steps and edges. By introducing an unsupervised geometric sensing module, high-precision reconstruction of the global and transient thermal field of key machine tool components is achieved, significantly reducing the dependence on the number and deployment location of sensors.

[0046] 2. This invention constructs a geometric prior-based residual sampling strategy (GPRS), which can dynamically generate heterogeneous sampling probability distributions based on geometric singularity distance fields and surface distance fields. In the early stages of training, this strategy utilizes geometric priors to avoid the "cold start" blind spot of traditional adaptive sampling. In the later stages of training, it automatically focuses on boundary layers and geometrically abrupt regions with drastic changes in thermal flux through residual feedback. This adaptive sampling mechanism effectively improves the model's ability to capture high-frequency thermal gradient features and achieves Pareto optimal allocation of computational resources in the spatiotemporal domain.

[0047] 3. This invention proposes a geometry-aware adaptive weight adjustment mechanism (G-SoftAdapt). Addressing the varying convergence difficulty across different regions of machine tool components (complex and smooth regions), a dynamic weight update model based on the loss change rate and geometric difficulty factor is established. This mechanism effectively resolves the imbalance in magnitude between partial differential equation residuals, boundary condition residuals, and sparse data constraints without introducing additional gradient computation overhead, significantly enhancing the model's convergence stability and prediction robustness when handling multi-scale geometric features and complex boundary conditions. Attached Figure Description

[0048] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the 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.

[0049] Figure 1 This is a flowchart of the machine tool thermal field reconstruction method based on geometric prior and dual adaptive PINN according to an embodiment of the present invention.

[0050] Figure 2 This is a schematic diagram of the sparse sensor layout in an embodiment of the present invention.

[0051] Figure 3 This is a comparison chart of the results of uniform sampling and adaptive sampling modules.

[0052] Figure 4 The training curves are for the baseline model.

[0053] Figure 5 The training curves for the GIDA model, which combines adaptive sampling and adaptive weights.

[0054] Figure 6 This is a reconstructed step-axis thermal field diagram of the GIDA model at the 2000th time step in the embodiment. Detailed Implementation

[0055] 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.

[0056] It should be noted that, unless otherwise specified, the features in the following embodiments and implementation methods can be combined with each other.

[0057] like Figure 1 As shown, a machine tool thermal field reconstruction method based on geometric prior and dual adaptive PINN includes the following steps:

[0058] (1) Arrange a small number of temperature sensors in key thermally sensitive parts of the machine tool components, and collect temperature data at a set frequency during the operation of the machine tool to obtain a sparse temperature dataset.

[0059] (2) Construct a geometric perception module to obtain uniformly dense candidate sampling point clouds from the three-dimensional model of the machine tool parts by dividing the volume mesh, and extract the surface point cloud set. and feature edge set The feature edges correspond to geometrically abrupt regions of the part (such as step edges or right-angled edges). For each candidate sampling point, the Euclidean distance to the nearest feature edge is calculated as the singularity distance field. And the Euclidean distance to the nearest surface as the surface distance field. Unsupervised region labeling is performed on candidate points to divide them into complex regions, surface regions, and internal regions.

[0060] (3) Construct an adaptive sampling module with geometric prior. During network training, a heterogeneous probability distribution is used to select training points from the candidate sampling point cloud. The sampling strategy includes a hot start phase and an adaptive phase, which use a pure geometric distribution and a residual hybrid distribution for dynamic sampling, respectively. The sampling probability distribution of the geometric prior is: boundary sampling probability and singularity distance field. Inversely proportional to the density of the step edges, the volume sampling probability follows a dual distance decay law, simultaneously affecting the surface distance field. and singularity distance field This is inversely proportional, allowing sampling points to automatically focus on areas of drastic heat flux change near the wall and behind geometrical abrupt changes. The residual sampling probability distribution is calculated from the candidate point residuals.

[0061] (4) Construct a geometry-aware adaptive weight module and assign a geometry difficulty factor to different regions using the region labels obtained in step (1). During training, the weights of each loss term are dynamically adjusted based on the historical rate of change of each loss component and the geometry difficulty factor. When training in a complex geometric region stalls, the algorithm automatically increases the weight priority of that region, and this process does not introduce additional gradient calculation overhead.

[0062] (5) A high-frequency enhanced physical information neural network structure was constructed to solve the transient heat conduction partial differential equation. The fully connected layers in the traditional PINN network structure are difficult to fit the high-frequency thermal characteristics at geometric abrupt changes. Therefore, a Gaussian Fourier Feature Embedding was introduced at the network input to map the normalized spatiotemporal coordinates to a high-dimensional feature space, forcing the network to focus on high-frequency temperature gradient changes. Adaptive sampling points and dynamic weights were used to jointly train the neural network. After training, the network was used to infer the temperature values ​​of machine tool components at any time and any location, completing the three-dimensional global thermal field reconstruction.

[0063] Specifically, the sparse temperature data in this embodiment of the invention is in the following form:

[0064] ;

[0065] In the formula, For sparse temperature datasets, For the number of temperature sensors, For the number of time sampling steps, Let i be the fixed three-dimensional spatial coordinates of the i-th sensor. For the j-th sampling time, Let be the temperature data collected by the i-th sensor at time j. This dataset will be reshaped before being input into the neural network. The input matrix (containing )and The label vector (containing) ),in This represents the total number of observed data points.

[0066] Specifically, the geometric sensing module in this embodiment of the invention is as follows: Figure 1 As shown in S1, the following steps are included:

[0067] S101. Construct candidate sampling point clouds and extract surface point clouds and feature edge sets.

[0068] In detail, the 3D model files of machine tool components (such as leadscrews and spindles) are read and tetrahedralized. All vertices of the mesh are extracted to form a surface point cloud set. Feature edge set The extraction is achieved by calculating the angle between the normal vectors of adjacent facets on the surface, and the calculation expression is as follows:

[0069] ;

[0070] In the formula, and For two adjacent surfaces and The normal vector, It is the angle between their normal vectors, if Greater than the preset threshold Then the vertices on the face will be assigned to the feature edge set. Meanwhile, in the continuous space within the model's bounding box... Built-in includes Candidate sampling point cloud of each point .

[0071] S102. Calculate the singularity distance field of the candidate points.

[0072] In detail, for any candidate point within the computational domain, the shortest Euclidean distance to the surface point cloud set and the feature edge set is calculated, and a singularity distance field is constructed. Its calculation expression is as follows:

[0073] ;

[0074] ;

[0075] In the formula, To compute any point in the candidate sampling point cloud within the domain, For surface point cloud collection A little bit above, For the set of characteristic edges A little bit above, For the Euclidean norm, The distance from the characterization point to the surface of the component. The distance from the representation point to the geometric abrupt change is used. To improve computational efficiency, this embodiment employs the KD-Tree algorithm for nearest neighbor search, reducing the time complexity from... Reduce to .

[0076] S103. Unsupervised region labeling of candidate points based on distance field.

[0077] In detail, based on the distance field values ​​calculated in step S102, a singularity distance threshold is set. and surface distance threshold Candidate points Divided into three mutually exclusive regions marked :

[0078] ;

[0079] In the formula, "Complex Region" represents a region with rich geometric features and drastic changes in thermal gradient (such as near steps or keyways); "Surface Region" represents a region directly affected by boundary conditions; and "Internal Region" represents the core region of the component where heat conduction is relatively stable. These region labels will be used for the allocation of geometric difficulty factors in the subsequent S3 step.

[0080] In detail, in most areas where sensor data is missing, especially in non-measuring areas inside machine tool components and on surfaces, dense physical sampling points generated based on the distance field are used to compensate. During model training, the network's output at these points is forced to conform to the physical equations, thus filling the information gaps. It should be noted that not all candidate points need to be input into the network for training; inputting all points would reduce training efficiency and convergence accuracy due to computational redundancy. Therefore, in subsequent steps, this invention will employ an adaptive sampling strategy to intelligently select key points from the candidate pool and automatically perform dense sampling in key areas such as geometrical abrupt changes, thereby achieving optimal allocation of computational resources.

[0081] Specifically, the adaptive sampling module for geometric priors in this embodiment of the invention is as follows: Figure 1 As shown in S2, it includes the following steps:

[0082] S201. Calculate the geometric prior sampling probability distribution based on the distance field.

[0083] In detail, the singularity range field calculated in S1 is used. and surface distance field Construct a heterogeneous geometric sampling probability density function. For the boundary computation domain... To ensure high-density coverage of geometrically abrupt changes such as steps and edges, a boundary sampling probability distribution is constructed for the sampling points. Its calculation expression is as follows:

[0084] ;

[0085] In the formula, Candidate points on the boundary, To prevent small constants from being divided by zero, To control the exponential hyperparameter of the degree of aggregation, The larger the value, the more concentrated the sampling points are at the points of geometric abrupt change. As a normalization constant, ensure .

[0086] For internal computing domain To ensure that the sampling points simultaneously focus on the near-wall thermal boundary layer and the region of drastic heat flux change behind the geometric abrupt change, a volumetric sampling probability distribution following a dual distance attenuation law is constructed, the calculation expression of which is:

[0087] ;

[0088] In the formula, As an internal candidate point, To prevent small constants from being divided by zero, , These values ​​control the degree to which sampling points in the volume domain cluster towards the surface and geometric abrupt changes; the larger the value, the more the sampling points cluster towards the surface and geometric abrupt changes. As a normalization constant, ensure .

[0089] S202. Calculate the residual sampling probability distribution based on network feedback.

[0090] In detail, during network training, the candidate pool is periodically... The point input in the current neural network calculates the absolute value of the residual of its partial differential equation. Based on this value, a residual sampling probability distribution is constructed, and its calculation expression is as follows:

[0091] ;

[0092] In the formula, For the candidate pool of sampling points, For a point in the candidate pool, The absolute value of the residual of the partial differential equation. The exponential hyperparameter used to adjust the residual sensitivity increases with the size of the residual probability distribution, which in turn affects the multitude sampling. This distribution significantly increases the probability that regions with large current prediction errors (i.e., regions where physical laws are not yet satisfied) will be selected in the next training round, with only one forward propagation and almost no additional computational overhead.

[0093] S203. Execute a phased dynamic hybrid sampling strategy.

[0094] In detail, in order to balance training efficiency and generalization ability, the training process is divided into a warm-up phase and an adaptive phase, with different sampling and weight coordination strategies adopted in different phases.

[0095] The hot start phase acts on the front training cycles ( The sampling points are generated based on the geometric prior distribution in S201, and the weights are set according to the geometric difficulty factor in S301. This static configuration based on strong geometric prior knowledge can force the model to prioritize learning the high-frequency hot features of geometric abrupt changes (such as edges and steps) in the early stage of training, thereby avoiding the divergence of the optimization direction due to the random fluctuation of the initial residuals and significantly accelerating the early convergence speed of the model.

[0096] The adaptive phase occurs after the warm-start phase, involving periodic resampling and dynamic weight updates. The sampling points are based on a hybrid probability distribution driven by geometric priors and residuals, and its calculation expression is as follows:

[0097]

[0098] In the formula, To compute any candidate point within the computation domain, The mixed sampling probability distribution function, Let be the geometric prior sampling probability distribution function, if For boundary sampling points, then ,like If the sampling point is a volume region, then it is , The residual probability distribution of the sampling points. and The weighting coefficients for the geometric probability distribution and the residual probability distribution are respectively, satisfying... This sampling strategy allows the model to focus on other regions within the computational domain (such as smooth transition zones or unknown hotspots) based on real-time error feedback. Simultaneously, the geometric weights in the mixed distribution... Furthermore, the geometric difficulty factor in the weight update formula ensures that even in the later stages of training, the physical features at geometrically abrupt changes are still prioritized and not overwhelmed or ignored by large areas of smooth data. In addition, this dynamic resampling based on error distribution introduces beneficial random perturbations and targeted constraints, achieving a similar effect to regularization, thereby effectively suppressing overfitting in the later stages of training.

[0099] Specifically, the adaptive weighting module for geometry perception in this embodiment of the invention is as follows: Figure 1 As shown in S3, it includes the following steps:

[0100] S301, Assign geometric difficulty factor based on region label.

[0101] In detail, based on the unsupervised region markers generated in step S103 For each loss item Assign the corresponding geometric difficulty factor This factor quantifies the difficulty of reconstructing the physical field within different geometric regions.

[0102] S302. Calculate the historical relative rate of change for each loss item.

[0103] In detail, in the first Epoch (the training period) is calculated. The historical relative rate of change of each loss term is calculated using the following expression:

[0104] ;

[0105] In the formula, Let k be the historical relative rate of change of the k-th loss term in the t-th training period. This represents the loss value for the current period. This is the baseline value for the average loss in the initial stage. This is the momentum coefficient. This index uses a moving average strategy to calculate the degree of convergence stagnation of the current loss term relative to other terms. The larger the value, the smaller the decrease of the loss term relative to the initial state (i.e., the slower or more difficult the convergence), and the higher the weight should be assigned.

[0106] S303, Update dynamic weights based on the Softmax mechanism.

[0107] In detail, the calculation expression for dynamic weight updates is as follows:

[0108] ;

[0109] In the formula, Let be the relative weight of the k-th loss term in the t-th training period. The total number of loss items. For smoothing coefficients, Let k be the historical relative rate of change of the k-th loss term. The geometric difficulty factor for the k-th loss term is assigned a larger value for complex regions. Sum the exponential values ​​of the weighted rates of change for all loss components.

[0110] This weight update mechanism has a dual adaptive characteristic: on the one hand, through... Automatically focus on the "weakest link" task that converges the slowest during training; on the other hand, through... This forces the network to continuously focus on the physical consistency of geometric abrupt changes throughout the entire training cycle, effectively preventing the model from overfitting in smooth regions and ignoring complex boundary features. Updated weights This will be directly applied to the joint loss function calculation in step S4.

[0111] Specifically, the PINN model construction and joint training phases in this embodiment of the invention are as follows: Figure 1 As shown in S4, it includes the following steps:

[0112] S401, Construct the Gaussian Fourier feature mapping module.

[0113] Specifically, addressing the "spectral bias" problem in standard multilayer perceptrons (MLPs) when fitting high-frequency thermal gradient features, this embodiment introduces Gaussian Fourier Feature Embedding at the network input. First, the spatiotemporal coordinates are input... After normalization, it is then mapped to a high-dimensional feature space, and its calculation expression is as follows:

[0114] ;

[0115] In the formula, The frequency matrix has the following elements. It follows an isotropic Gaussian distribution , The range of spectrum that the control network can perceive. The mapped feature vector. Replace the original coordinates As input to subsequent fully connected layers.

[0116] S402. Construct the heat conduction control equations and complex boundary condition constraints.

[0117] In detail, the network output temperature is calculated using automatic differentiation technology. Partial derivatives with respect to spacetime coordinates, residuals of transient heat conduction partial differential equations (PDEs) for computer tool components Its calculation expression is:

[0118] ;

[0119] In the formula, To compute the spatiotemporal coordinate vector within the computational domain, For material density, For specific heat capacity, Thermal conductivity, The gradient operator, when computed in a three-dimensional Cartesian coordinate system, represents vector differentiation. , The Laplacian operator represents the divergence of the gradient, specifically calculated as the sum of the second-order partial derivatives of temperature with respect to the three spatial coordinates.

[0120] ;

[0121] Simultaneously construct boundary condition residuals This embodiment sets two types of boundary conditions: convective heat transfer boundary conditions (Robin conditions) and surface heat flux input boundary conditions (Neumann conditions). The calculation expressions for the two types of boundary conditions are as follows:

[0122] ;

[0123] ;

[0124] In the formula, The formula for calculating the residuals of the convective heat transfer boundary condition (Robin condition) is as follows: The formula for calculating the residuals of the surface heat flow input boundary condition (Neumann condition) is as follows: For sampling points on the convection boundary, Let be the surface normal vector at that point. The convective heat transfer coefficient is... For ambient temperature, Sampling points on the heat flux boundary, The input is the heat flux density.

[0125] S403. Construct a joint loss function and perform optimization training.

[0126] In detail, by integrating PDE physical constraints, mixed boundary condition constraints, and sparse data constraints, a joint loss function is constructed, and its calculation expression is as follows:

[0127] ;

[0128] In the formula, For PDE sampling domain, and Let be the set of sampling points and the dynamic weights of the k-th PDE loss term, respectively. For PDE residuals, For the boundary condition sampling domain, and Let be the set of sampling points and the dynamic weights of the b-th boundary condition loss term, respectively. For boundary condition residuals, For the weights of the data loss items, For sparse sensor numbers, and Let be the predicted temperature and the actual temperature at the i-th sensor location, respectively. This represents the mean square error.

[0129] Based on the total loss function The Adam optimizer is used to iteratively update the neural network parameters. After training, the network can be used to infer the temperature field distribution of machine tool components at any time and any location, thus completing the three-dimensional thermal field reconstruction.

[0130] To verify the feasibility of the machine tool thermal field reconstruction method based on geometric prior and dual adaptive PINN proposed in this invention, a typical machine tool stepped shaft model was constructed for experimental verification in this embodiment. The stepped shaft is made of 45# steel, and its geometric dimensions consist of three cylindrical segments, with dimensions (diameter × length) from left to right as follows: (50mm, 150mm), (100mm, 700mm), (50mm, 150mm).

[0131] The experimental data were obtained as follows: A three-dimensional model of the stepped shaft was established in finite element simulation software. The initial temperature was set to the ambient temperature, and a continuous heat flow input was applied to both end faces of the shaft as boundary conditions. A transient heat conduction simulation with a total duration of 2000 time steps was performed. The full-field temperature data was output every 10 time steps, resulting in 201 temperature field snapshots at various time sections (including time t=0). The stepped shaft model has a total of 8482 mesh vertices, therefore the total sample data is 8482 × 201 = 1704882. These data are used as "Ground Truth" to verify the accuracy of the model reconstruction.

[0132] To simulate the sparse monitoring environment of an industrial site, this embodiment selects only 4 points each on the two end edges of the stepped shaft and the surfaces of the two thin shafts (50mm diameter segments), arranging a total of 16 virtual temperature sensors. Figure 2As shown, only 16×201=3216 temperature data points corresponding to these 16 locations are extracted as a sparse training set to drive the training of the PINN model. The temperature data of all other grid points are not visible and are only used for the final verification test.

[0133] In detail, the effectiveness of the proposed Geometric Prior Adaptive Sampling Strategy (GPRS) is verified, such as... Figure 3 As shown, the distribution of sampling points on the stepped axis is compared between the traditional uniform sampling strategy and the GPRS adaptive sampling strategy of this invention. The figure shows that under the uniform sampling strategy, the sampling points are randomly scattered throughout the computational domain, while under the GPRS strategy, the sampling points are located in the geometrically singular distance field. Smaller regions (i.e., the step connections of the stepped axis, right-angled edges) and surface distance fields Smaller regions (i.e., the boundary layer) exhibit significant high-density clustering. This distribution pattern aligns with the drastic changes in heat flux density at geometric abrupt changes in the physical mechanism of heat conduction, demonstrating that the GPRS strategy can effectively guide the network to focus on high-frequency thermal characteristic regions.

[0134] In detail, the impact of the dual adaptive framework on model training convergence is verified. Figure 4 and Figure 5 The training loss curves of the baseline PINN model (without geometric prior and adaptive mechanism) and the GIDA model proposed in this invention (based on geometric prior and dual adaptive PINN) are compared. As shown in the figure, the GIDA model's loss decrease curve reaches the vicinity of the optimum faster than the baseline model, with a convergence speed improvement of approximately 50%. Furthermore, the severe overfitting phenomenon that occurs in the later stages of training in the baseline model is effectively suppressed in the GIDA model.

[0135] In detail, the model's performance in 3D thermal field reconstruction under sparse data was verified. Using only 0.1% of the data for training, the GIDA model achieved MAE and RMSE errors of 0.7204 and 1.2211 respectively in the full-field prediction results, with relative errors and L2 relative errors of 3.02% and 4.78%, respectively. These represent improvements of 54.2% and 39.92% compared to the baseline model. Figure 6 The results demonstrate the 3D transient thermal field reconstruction of the GIDA model at the 2000th time step. This proves the superiority of the method of this invention in accurately reconstructing the 3D transient thermal field of machine tool components using sparse data.

[0136] The embodiments described above provide a detailed explanation of the technical solutions and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A machine tool thermal field reconstruction method based on geometric prior and double adaptive PINN, characterized in that, Includes the following steps: (1) Arrange several temperature sensors on the thermally sensitive parts of the machine tool components and collect sparse temperature data during the operation of the machine tool; (2) Construct a geometric perception module to build a uniformly dense candidate sampling point cloud from the three-dimensional model of the machine tool component; Extract the surface point cloud set and feature edge set, calculate the singularity distance field and surface distance field of each candidate sampling point, and perform region labeling; The singularity distance field is the Euclidean distance from the sampling point to the nearest feature edge, and the surface distance field is the shortest Euclidean distance from the sampling point to the surface of the machine tool component; (3) Construct an adaptive sampling module with geometric priors to dynamically select training points from the candidate sampling point cloud; specifically including: During the warm-up phase of training, sampling is performed solely based on the geometric sampling probability distribution, which includes the boundary sampling probability distribution and the volume sampling probability distribution. For sampling points in the boundary computational domain, a boundary sampling probability distribution is constructed, and the boundary sampling probability is inversely proportional to the singularity distance field. For sampling points in the internal computational domain, a volume sampling probability distribution is constructed, and the volume sampling probability follows a dual distance decay law, being inversely proportional to both the surface distance field and the singularity distance field. The adaptive training phase employs a hybrid sampling method combining geometric sampling probability distribution and residual sampling probability distribution. Points from the candidate sampling point cloud are periodically input into the current PINN model to calculate the absolute value of the residual of its partial differential equation, and a residual sampling probability distribution is constructed based on this value. (4) Construct a geometrically-aware adaptive weight module to assign geometric difficulty factors to different regions based on region labels. During training, the weights of each loss term are dynamically adjusted based on the historical rate of change of each loss term combined with the geometric difficulty factor. (5) Construct the PINN model with Gaussian Fourier feature mapping, use the sparse temperature dataset in step (1) and the training points in step (3) to construct the joint loss function, and train the PINN model by combining the adaptive sampling module and the adaptive weight module. (6) After training, the temperature value of the machine tool component at any time and any position is inferred using the PINN model to complete the three-dimensional global thermal field reconstruction.

2. The machine tool thermal field reconstruction method based on geometric prior and dual adaptive PINN according to claim 1, characterized in that, In step (2), each candidate point is labeled as a different region, including a complex region, a surface region, and an interior region, by setting a singularity distance threshold and a surface distance threshold .

3. The machine tool thermal field reconstruction method based on geometric prior and dual adaptive PINN according to claim 1, characterized in that, In step (5), a joint loss function is constructed by integrating physical constraints, boundary condition constraints, and sparse temperature data constraints.

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