Method for predicting temperature field of machine tool components based on spectral geometry physical information neural network

Abstract

The application discloses a machine tool component temperature field prediction method based on a spectral geometry physical information neural network, and comprises the following steps: obtaining a three-dimensional geometric model of a machine tool component whose temperature field is to be reconstructed, performing discretization processing to obtain mesh data containing a node set and a connection relationship between nodes; constructing a discrete Laplace-Beltrami operator and extracting spectral geometry embedding features of each node; constructing a physical information neural network model; constructing a composite loss function containing a boundary condition loss, a physical equation loss and a data-driven loss; training the physical information neural network; inputting the spectral geometry embedding features of a query point into the trained physical information neural network, calculating and outputting the reconstructed temperature value of the point in a three-dimensional space, and obtaining a full-field three-dimensional thermal distribution. By using the application, the solving speed and prediction accuracy of the temperature field prediction of key machine tool components can be improved.

Description

Technical Field

[0001] This invention belongs to the field of precision machining and CNC machine tool technology, and in particular relates to a method for predicting the temperature field of machine tool components based on a spectral geometric physical information neural network. Background Technology

[0002] In the field of modern high-end equipment manufacturing, the machining accuracy of CNC machine tools directly determines the manufacturing quality of complex structural parts. During the actual operation of machine tools, thermal errors caused by internal heat sources (such as motors, bearings, and cutting heat) and external environmental temperature fluctuations have become the primary factor restricting the improvement of their accuracy, accounting for 40% to 70% of the total geometric error of machine tools. Therefore, in-depth research on temperature field prediction methods for key machine tool components, and the achievement of accurate prediction of the temperature field of key machine tool components, is of great significance for improving the machining accuracy, enhancing machining quality, and extending equipment life.

[0003] Traditional analysis methods primarily rely on finite element method (FEM) simulation. For example, Chinese patent document CN105022344A discloses a method for thermal error compensation in CNC machine tools. Based on finite element simulation analysis and using particle swarm optimization algorithm, it obtains key temperature points for thermal error compensation, and uses the temperature measurements at these key temperature points as the basis for thermal error compensation. Chinese patent document CN103273380A discloses an online monitoring method for thermal deformation error in CNC machine tools based on a simulation model. This method combines the finite element model of the CNC machine tool with actual measurement experiments to solve the problem of online monitoring of thermal deformation error in CNC machine tools.

[0004] Although thermal simulation methods are highly accurate, they have the following significant drawbacks: poor structural adaptability and strong mesh dependence. For structures with complex geometry and topology, generating high-quality body-fitting meshes is extremely time-consuming and difficult, and poor meshes can lead to non-convergence of calculations. They also have low computational efficiency. When solving inverse problems (such as heat source inversion) or involving design optimization, repeated iterative calculations are required, resulting in a large amount of computation and making real-time prediction difficult.

[0005] In recent years, deep learning methods based on Physical Information Neural Networks (PINNs) have attracted attention due to their "meshless" characteristics. However, existing PINN methods face the following problems when dealing with complex geometries: 1. Poor adaptability to complex topologies. Machine tool rotary tables typically contain complex non-convex geometries such as stiffeners, cooling channels, and bolt holes. Traditional PINNs directly use Euclidean coordinates. As input, neural networks struggle to perceive these complex geometric topological features. Two points that are spatially close may be far apart on the heat conduction path, leading to low prediction accuracy at geometrically abrupt changes. Secondly, there is the issue of spectral bias. Conventional fully connected neural networks tend to prioritize learning low-frequency functions, making it difficult to capture high-frequency details in the temperature field (such as drastic temperature rises near heat sources). On complex three-dimensional manifolds, simple coordinate mapping results in extremely slow or even non-convergent network convergence.

[0006] Therefore, there is an urgent need for an efficient thermal field reconstruction method that can sense complex geometric topologies and overcome spectral biases. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention provides a method for predicting the temperature field of machine tool components based on a spectral geometric physical information neural network. This method solves the problems of low accuracy, slow convergence, and spectral deviation in complex topological structures of traditional methods, and improves the solution speed and prediction accuracy of temperature field prediction for key machine tool components (such as rotary tables, electric spindles, and lead screws).

[0008] A method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network includes the following steps:

[0009] (1) Obtain the three-dimensional geometric model of the temperature field of the machine tool component to be reconstructed, and discretize the three-dimensional geometric model to obtain mesh data containing the set of nodes and the connection relationship between nodes;

[0010] (2) Construct a discrete Laplace-Beltrami operator based on grid data and extract the spectral geometric embedding features of each node;

[0011] (3) Construct a physical information neural network model, whose input layer receives spectral geometric embedding features and whose output layer outputs the temperature prediction value of the corresponding node;

[0012] (4) Construct a composite loss function that includes boundary condition loss, physical equation loss and data-driven loss;

[0013] (5) The physical information neural network is trained using a gradient-based optimization algorithm. The network parameters are iteratively updated by minimizing the composite loss function until the model converges.

[0014] (6) Using the trained physical information neural network, input the spectral geometric embedding features of the point to be queried, calculate and output the reconstructed temperature value of the point in three-dimensional space, and thus obtain the three-dimensional heat distribution of the whole field.

[0015] This invention enhances the model's ability to perceive geometric connectivity by introducing spectral geometric embedding features based on the Laplace-Beltrami operator to map low-dimensional Euclidean coordinates to a high-dimensional manifold feature space. Simultaneously, by combining physical constraints based on the discrete Laplace energy form, the Dirichlet energy is directly calculated using sparse matrix operations, avoiding the expensive second-order automatic differentiation calculations performed on the neural network in traditional methods, significantly reducing computational complexity and memory usage. Furthermore, by employing a two-stage optimization strategy combining Adam and L-BFGS, both global convergence and local precision of training are considered, thereby achieving efficient and high-precision reconstruction of the three-dimensional thermal field of industrial structures with complex geometric features.

[0016] In step (1), the three-dimensional geometric model is discretized into a discretized mesh model containing multiple nodes using a mesh generation tool. Its definition is:

[0017] ;

[0018] In the formula, Indicates inclusion A set of vertices of nodes. Indicates inclusion A set of triangular facet units; where any node All correspond to three-dimensional Euclidean space A coordinate vector Its formula is:

[0019] .

[0020] The specific process of step (2) is as follows:

[0021] Constructing a stiffness matrix based on grid data With the mass matrix Solve the generalized eigenvalue problem;

[0022] After solving, select the first The eigenvectors corresponding to the non-zero minimum eigenvalues ​​are used as the spectral geometric embedding features of each node.

[0023] Preferably, The range of values ​​is .

[0024] Furthermore, the stiffness matrix The cotangent weight matrix is ​​a A sparse symmetric matrix of dimension 1, whose element values ​​reflect the geometric connection strength between grid nodes.

[0025] Calculating sparse symmetric matrices using the cotangent weighting method The specific process is as follows:

[0026] For non-diagonal elements (when node) and When adjacent), the formula is:

[0027] ;

[0028] In the formula and These are the two vertex angles on the mesh surface that are opposite to the common edge.

[0029] For diagonal elements Its formula is:

[0030] ;

[0031] In the formula Represents a node The set of all first-order neighbor nodes.

[0032] For non-adjacent nodes, the formula is:

[0033] .

[0034] To address the node density variation issue caused by non-uniform meshes, a mass matrix is ​​introduced. Normalization is performed, and for computational efficiency, it is preferable to calculate the lumped mass matrix, which is usually a diagonal matrix used to describe the size of the geometric region controlled by each node. This is used to correct for uneven grid sampling density, and its formula is:

[0035] ;

[0036] In the formula, For the mass matrix The first on the diagonal There are 0 elements (all non-diagonal elements are 0). The area of ​​the patch. For all nodes The set of triangular facet units with vertices.

[0037] The problem to be solved is the generalized eigenvalue problem: By solving this equation, the feature values ​​used as inputs to the neural network can be obtained. and eigenvectors (Spectral geometric features).

[0038] In step (3), a physical information neural network model is constructed, and the input layer receives the k-dimensional spectral geometric embedding features generated in step (2). Assume the network contains... Layer, number Layer output The calculation formula is:

[0039] ;

[0040] In the formula and These are the weight matrix and bias vector of the l-th layer, respectively. The activation function (tanh is used in this method) is used. The input features are used as inputs. The output layer outputs normalized temperature predictions. The formula is:

[0041] .

[0042] In step (4), the composite loss function is defined as:

[0043] ;

[0044] in, This is a boundary condition loss used to constrain convective heat transfer on the surface of the component; The loss is due to the physical equations and is used to constrain the temperature distribution across the entire field to satisfy the heat conduction equations. Data-driven loss; , , These are the weighting coefficients for each loss term.

[0045] Boundary condition loss Considering the root mean square error constraint of the heat flux balance on the component surface, including the thermal conductivity and convective heat flux of the component, the formula is:

[0046] ;

[0047] in, The rate at which heat is supplied from the interior to the surface; The heat transfer intensity ratio is determined by the convective heat transfer coefficient. With the thermal conductivity of the material The ratio constitutes; The ambient temperature.

[0048] Physical equation loss Employing a discrete energy norm-based approach The form is constructed as follows:

[0049] ;

[0050] in, This represents the temperature vector of all grid nodes output by the fully connected neural network. Here is the stiffness matrix; Given the known heat source distribution vector, this formula directly constrains the heat flow balance across the entire field through matrix operations.

[0051] Data-driven loss This is the mean square error between the measured temperature from the sampling temperature sensor and the predicted temperature at the corresponding node. The formula is:

[0052] ;

[0053] In the formula, This refers to the set of grid node indices selected based on the sampling locations at the sensor interface, i.e., the set of all grid nodes located within the observation interface area. The physical information neural network is for the first The normalized temperature prediction value output by each node. Indicates the first The known normalized temperature observations corresponding to each node in the actual physical system are the real temperature data used as supervision labels.

[0054] In step (5), the gradient-based optimization algorithm includes a two-stage optimization strategy: In the first stage, the first-order adaptive moment estimation optimization algorithm (Adam) is used, and a first preset number of iterations is set to perform global preheating training on the physical information neural network so that the model can quickly adapt to the boundary condition data; In the second stage, the second-order quasi-Newton optimization algorithm (L-BFGS) is used, and a second preset number of iterations is set to perform local fine training using the inverse matrix approximation information of the Hessian matrix until the composite loss function converges to the preset threshold.

[0055] The iterative update process for optimizing the parameters of a physical information neural network is to minimize the total loss function. To achieve the goal, find the optimal network parameters. The formula is:

[0056] ;

[0057] In the first stage, the Adam optimizer is used to update the parameters based on the first-order gradient information; in the second stage, the L-BFGS optimizer is used to approximate the inverse of the Hessian matrix based on the quasi-Newton method. Perform parameter updates, and the direction of those updates. The formula is:

[0058] ;

[0059] in, For the first The physical information neural network model parameters (including weight matrix and bias vector) at the next iteration.

[0060] Compared with the prior art, the present invention has the following beneficial effects:

[0061] 1. This invention features high-precision geometric adaptability. By utilizing the spectral geometry embedding method, the network can "understand" the connectivity and topology of complex structures, avoiding thermal prediction errors in non-connected regions. It is particularly suitable for complex structures with dense fins, such as lead screws and cooling channels in key machine tool components.

[0062] 2. This invention has a significant computational acceleration effect. Compared with traditional CFD / FEM, the model trained by this invention can achieve millisecond-level inference; compared with ordinary PINN, spectral embedding accelerates the learning of high-frequency information and is more suitable for complex structural parts than traditional coordinate input; relying on physical equation constraints, only a few measurement points on the surface are needed to deduce the internal temperature. Attached Figure Description

[0063] 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.

[0064] Figure 1 This is a schematic diagram of the process for predicting the temperature field of machine tool components based on a spectral geometric physical information neural network, according to an embodiment of the present invention.

[0065] Figure 2 This is a histogram of the predicted temperature error in an embodiment of the present invention.

[0066] Figure 3 The prediction result R in the embodiment of the present invention 2 Fit a scatter plot. Detailed Implementation

[0067] This embodiment focuses on temperature field prediction for a typical machine tool key component, a hollow lead screw. The following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be noted that the embodiments described below are intended to facilitate understanding of the present invention and do not limit it in any way.

[0068] like Figure 1 As shown, the method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network includes the following steps:

[0069] Step S1: Hollow screw modeling and data preprocessing.

[0070] A 3D geometric model of the hollow screw to be analyzed was generated using CAD modeling tools, and then discretized using a mesh generation tool, containing a set of nodes. and tetrahedral unit set The model surface is discretized into a subset containing temperature observations of some known nodes. vertices and A triangular mesh of individual facets. To accelerate neural network convergence and eliminate dimensional effects, temperature... Normalization is performed.

[0071] Step S2: Spectral geometry embedding calculation.

[0072] Stiffness matrix calculated based on mesh topology With the mass matrix Subsequently, the following generalized eigenvalue problem is solved to extract spectral features, and its formula is:

[0073] ;

[0074] Select the first corresponding feature value Construct the spectral feature matrix from the eigenvectors. For any node Its input feature vector is no longer Instead, it is a high-dimensional spectral eigenvector, and its formula is:

[0075] .

[0076] Step S3: Building the physical information neural network architecture.

[0077] The network was trained using the Physical Information Neural Network (PINN) framework. The neural network architecture includes a spectral feature fusion layer, a deep fully connected layer, and a physically constrained output layer.

[0078] Construct a fully connected neural network (MLP). Number of input layer nodes: 128 (corresponding to spectral feature dimension). Hidden layers: 4 layers, each containing 128 neurons, using the hyperbolic tangent function tanh as the activation function, and initializing the weights using a Xavier Uniform distribution; Output layer: 1 node (outputs a normalized scalar temperature prediction value). This layer uses a no-activation-function (linear output) setting.

[0079] Step S4: Define the composite physical loss function.

[0080] To reconstruct a physically consistent temperature field in an unlabeled region, a composite loss function is defined, comprising three parts: boundary error, physical constraints, and data constraints.

[0081]

[0082] In this embodiment, the weights are configured as follows: , For boundary condition loss, For the loss of discrete physical equations, Data-driven loss.

[0083] As a boundary condition loss, a mean square error is used to constrain known points. By minimizing this loss term, it is ensured that the reconstructed temperature field strictly follows Newton's law of cooling in the sensorless surface region. The formula is as follows:

[0084] ;

[0085] In the formula, The rate at which heat is supplied from the interior to the surface; The heat transfer intensity ratio is determined by the convective heat transfer coefficient. With the thermal conductivity of the material The ratio constitutes; The ambient temperature.

[0086] To account for the loss in discrete physical equations, for the steady-state heat conduction equation The stiffness matrix obtained by step S2 With the mass matrix The discrete form of the Dirichlet energy expression for heat conduction across the entire field is directly calculated, avoiding automatic differentiation calculations. The formula is as follows:

[0087] ;

[0088] in, This represents the temperature vector of all grid nodes output by the fully connected neural network. Here is the stiffness matrix; The heat source distribution vector is known.

[0089] The data-driven loss error term is formulated as follows:

[0090] ;

[0091] In the formula, This refers to the set of grid node indices selected based on the sampling locations at the sensor interface, i.e., the set of all grid nodes located within the observation interface area. The physical information neural network is for the first The normalized temperature prediction value output by each node. Indicates the first The known normalized temperature observations corresponding to each node in the actual physical system are the real temperature data used as supervision labels.

[0092] Step S5: Two-stage optimization strategy for network training.

[0093] A two-stage training strategy of "global approximation + local fine-tuning" is adopted to solve the problem of insufficient accuracy or easy trapping in local optima by a single optimizer.

[0094] Phase 1 (Adam Global Optimization): Train the Adam optimizer for 10,000 epochs with an initial learning rate of 0.001. A warmup period of 1,000 epochs is introduced, during which only computation is performed. ,make =0 enables the network to quickly adapt to boundary data and establish a preliminary temperature profile.

[0095] The second stage (L-BFGS local fine-tuning): After the loss decreases gradually, switch to the L-BFGS second-order optimizer, set the maximum number of iterations to 2000, and adopt the Strong Wolfe Line Search strategy. Use the second derivative approximation information of the Hessian matrix to achieve high-precision convergence and significantly reduce the residual.

[0096] Step S6: Result Evaluation.

[0097] The prediction results of the trained model are compared with the calculation results of the commercial software Abaqus. The error histogram of the comparison results is shown below. Figure 2 As shown, this indicates that most of the error is concentrated below 1℃, R 2 Fitted scatter plot as follows Figure 3 As shown, its value is 0.9997, indicating a good prediction effect. The results show that this method can clearly reconstruct the temperature field distribution of the lead screw.

[0098] The comparative experiment mainly compares the accuracy of the finite element method, the commonly used PINN method, and the method of this invention in predicting the temperature field of the same component.

[0099] (a) Finite Element Method (FEM): Steady-state thermal analysis of hollow screws is performed in Abaqus, with boundary conditions consistent with convective heat exchange conditions, and the temperature field results obtained are used as a standard accuracy reference.

[0100] (b) Traditional PINN: A standard PINN method with pure three-dimensional coordinate input. The network structure is consistent with that of this method, and the boundary conditions for convection and heat exchange are also consistent.

[0101] (c) The method of the present invention: namely, the temperature field prediction method based on the spectral geometric input physical information neural network, while the boundary conditions and convective heat exchange conditions remain consistent.

[0102] The prediction accuracy of each method for hollow screws is shown in Table 1.

[0103] Table 1

[0104]

[0105] The comparative results show that the proposed method has a higher training speed than the traditional coordinate input-based PINN, and also has higher accuracy and less fluctuation in solution accuracy. This demonstrates the limitations of the traditional PINN in predicting the temperature of complex machine tool components, indicating that the proposed method is more suitable for predicting the temperature of complex machine tool components.

[0106] 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 method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network, characterized in that, Includes the following steps: (1) Obtain the three-dimensional geometric model of the temperature field of the machine tool component to be reconstructed, and discretize the three-dimensional geometric model to obtain mesh data containing the set of nodes and the connection relationship between nodes; (2) Construct a discrete Laplace-Beltrami operator based on grid data and extract the spectral geometric embedding features of each node; (3) Construct a physical information neural network model, whose input layer receives spectral geometric embedding features and whose output layer outputs the temperature prediction value of the corresponding node; (4) Construct a composite loss function that includes boundary condition loss, physical equation loss and data-driven loss; Boundary condition loss Considering the root mean square error constraint of the heat flux balance on the component surface, including the thermal conductivity and convective heat flux of the component, the formula is: ； in The rate at which heat is supplied from the interior to the surface; The heat transfer intensity ratio is determined by the convective heat transfer coefficient. With the thermal conductivity of the material The ratio constitutes; The ambient temperature; Physical equation loss Employing a discrete energy norm-based approach The form is constructed as follows: ； in, This represents the temperature vector of all grid nodes output by the fully connected neural network. Here is the stiffness matrix; Given the known heat source distribution vector; Data-driven loss This is the mean square error between the measured temperature of the sampling temperature sensor and the predicted temperature of the corresponding node. (5) The physical information neural network is trained using a gradient-based optimization algorithm. The network parameters are iteratively updated by minimizing the composite loss function until the model converges. (6) Using the trained physical information neural network, input the spectral geometric embedding features of the point to be queried, calculate and output the reconstructed temperature value of the point in three-dimensional space, and thus obtain the three-dimensional heat distribution of the whole field.

2. The method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network according to claim 1, characterized in that, The specific process of step (2) is as follows: Constructing a stiffness matrix based on grid data With the mass matrix Solve the generalized eigenvalue problem; After solving, select the first The eigenvectors corresponding to the non-zero minimum eigenvalues ​​are used as the spectral geometric embedding features of each node.

3. The method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network according to claim 2, characterized in that, The range of values ​​is .

4. The method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network according to claim 2, characterized in that, Stiffness matrix The cotangent weight matrix is ​​a A sparse symmetric matrix of dimension 1, whose element values ​​reflect the geometric connection strength between grid nodes.

5. The method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network according to claim 2, characterized in that, In step (4), the composite loss function is defined as: ； in, This is a boundary condition loss used to constrain convective heat transfer on the surface of the component; The loss is due to the physical equations and is used to constrain the temperature distribution across the entire field to satisfy the heat conduction equations. Data-driven loss; , , These are the weighting coefficients for each loss term.

6. The method for predicting the temperature field of machine tool components based on a spectral geometrical physical information neural network according to claim 1, characterized in that, In step (5), the gradient-based optimization algorithm includes a two-stage optimization strategy: In the first stage, a first-order adaptive moment estimation optimization algorithm is adopted, and a first preset number of iterations is set to perform global preheating training on the physical information neural network so that the model can quickly adapt to the boundary condition data; In the second stage, a second-order quasi-Newton optimization algorithm is adopted, and a second preset number of iterations is set to perform local fine training using the inverse matrix approximation information of the Hessian matrix until the composite loss function converges to the preset threshold.

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