Fluid radial sliding bearing thermal performance prediction method based on physical information network

Abstract

The invention relates to the technical field of bearing thermal performance analysis, in particular to a fluid radial sliding bearing thermal performance prediction method based on a physical information network. And embedding the coupled Reynolds equation and energy equation residual error into a loss function, and combining a hard constraint boundary, a dynamic loss weight, a hybrid optimizer and a transfer learning strategy to realize rapid and high-precision prediction of the bearing multi-physics field. The method does not need to depend on a pre-generated large-scale simulation data set, the single prediction time can reach the millisecond magnitude after training is completed, and the method has extremely high physical consistency and generalization ability, can be quickly expanded to different bearing structure parameters and operation working conditions, and provides an efficient tool for bearing optimization design and real-time analysis.

Description

Technical Field

[0001] This invention relates to the field of bearing thermodynamic performance analysis technology, specifically a method for predicting the thermodynamic performance of fluid radial sliding bearings based on physical information networks. Background Technology

[0002] In high-precision, high-speed rotating machinery, fluid radial sliding bearings are widely used in aerospace, micro-mechanical systems, and precision instruments. Their operating state is affected by the coupling of multiple physical mechanisms, including flow, heat conduction, and structural deformation. Accurately modeling the complex multi-physical processes inside the bearing and predicting its thermodynamic performance are crucial prerequisites for ensuring system stability and reliability.

[0003] Currently, the analysis of bearing lubrication characteristics mainly relies on traditional numerical methods such as the finite difference method (FDM), the finite element method (FEM), or the finite volume method (FVM). Although these methods can achieve high computational accuracy under specific working conditions, they suffer from problems such as cumbersome model construction, high computational cost, and limited generalization ability. In particular, when dealing with strongly nonlinear and multi-physics coupled problems, it is difficult to balance accuracy and efficiency.

[0004] To overcome the efficiency bottleneck of traditional numerical methods, reduced-order intelligent models (surrogate models) based on simulation data-driven approaches have emerged. However, the prediction accuracy of such data-driven methods is highly dependent on large-scale, high-fidelity training datasets, the generation of which requires time-consuming traditional numerical simulations. Furthermore, the "black box" nature of the models leads to poor physical interpretability, and the prediction results may lack physical consistency. Summary of the Invention

[0005] The purpose of this invention is to provide a method for predicting the thermodynamic performance of fluid radial sliding bearings based on physical information networks. This method can balance computational efficiency, prediction accuracy, physical consistency, and generalization ability, thereby overcoming the shortcomings of existing technologies.

[0006] To achieve the above objectives, the present invention provides the following technical solution: a method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network, comprising the following steps: S1. Construct a mathematical model characterizing the thermo-mechanical coupling characteristics of a fluid radial sliding bearing. The mathematical model includes the Reynolds equation, the two-dimensional steady-state energy equation, the lubricating film thickness equation, and the velocity field equation, and performs dimensionless processing on each equation. S2. Design a multi-network parallel physical information network (PINN) architecture, including at least two fully connected neural networks, which are used to predict the pressure field and temperature field of the lubricating film, respectively. The physical equations are coupled between the networks through parameter sharing. S3. Construct a composite loss function that includes the residuals of partial differential equations (PDEs), boundary losses, and parameter penalty terms. Hard constraints are used to force the axial boundary conditions to converge to the given boundary values, while soft constraints are used to handle the circumferential periodic boundaries. S4. A hybrid placement point generation strategy is adopted, which includes: generating fixed static placement points through Latin hypercube sampling; adaptively updating dynamic placement points based on gradient change indicators during training; and using both static and dynamic placement points as training inputs. S5. Design a dynamic loss weight adjustment strategy based on uncertainty measurement to automatically quantify the confidence level of each loss item and assign weights. S6. A training strategy using dual optimizers and adaptive annealing learning rate is adopted. First, the Adam optimizer is used for global search, and then the L-BFGS optimizer is switched to perform fine optimization. S7. Combining transfer learning strategies, the pre-trained model is extended to new bearing structural parameters or operating conditions through gradient descent, enabling rapid prediction of thermal performance.

[0007] Preferably, the Reynolds equation is used to characterize the generalized two-dimensional, variable viscosity, compressibility, and steady-state pressure distribution of the lubricating film in radial sliding bearings, considering fluid compressibility (for gas bearings) or cavitation effects (for oil film bearings) and viscosity-temperature effects. The equation is as follows: (1) in, For the pressure distribution of the lubricating film, These are the circumferential, film thickness, and axial coordinates, respectively. It refers to the thickness of the lubricating film. It refers to the viscosity of the lubricating film, which is related to the temperature of the lubricating film. It is the journal speed. It is the journal rotational angular velocity. It is the bearing radius; The expression for the lubricating film thickness equation is as follows: (2); in It is the radial clearance of the bearing. It is the bearing eccentricity; The two-dimensional steady-state energy equation neglects heat transfer and convection changes in temperature along the film thickness direction, and its expression is: (4) in, It is the two-dimensional temperature field distribution of the lubricating film. It is air density. It is specific heat capacity. It is the thermal conductivity, and the velocity field equations include the circumferential and axial velocity components, respectively. and The expression is obtained by solving for the pressure field gradient of the lubricating film, as shown below: ； (3).

[0008] Preferably, the dimensionless processing involves normalizing physical quantities in the coupled mathematical model, such as lubricating film pressure, temperature, thickness, and bearing size, which have significant differences in magnitude, so that the dimensionless values ​​of each physical quantity are within the same order of magnitude. The resulting dimensionless equation is: (5) In the formula, , and The coordinates are dimensionless. , , , and The pressure, temperature, velocity components, and lubricating film thickness are dimensionless. , - These are dimensionless coefficients. This refers to the axial length of the bearing.

[0009] Preferably, the multi-network parallel Physical Information Neural Network (PINN) architecture satisfies: The temperature field output by the temperature prediction network is used to update the viscosity of the lubricating film, and then substituted into the Reynolds equation to calculate the residuals of the partial differential equation (PDE). The pressure field output by the pressure prediction network is used to solve the velocity field, and the density is updated in combination with the ideal gas law to correct the convection and dissipation terms in the energy equation. The network is based on a multilayer perceptron and introduces a self-attention mechanism. It generates diverse attention weights through two sets of encoders and dynamically weights and combines the features of each layer. The Tanh activation function is used to achieve smooth nonlinear mapping and alleviate the gradient vanishing problem; The weights are initialized using the Xavier normal distribution to balance the variance of the input and output layers and ensure effective gradient propagation.

[0010] Preferably, the axial boundary conditions include axial end face boundaries and circumferential periodic boundaries, as follows: Axial end face boundary: The dimensionless lubrication film pressure satisfies Temperature meets ; Circumferential periodic boundary: circumferential starting position after dimensionless transformation and termination position At this point, the physical quantity's numerical value and its first derivative are all equal, as expressed below: ; (6).

[0011] Preferably, the hard constraint method forces the Reynolds equation and the energy equation to converge to a given boundary value at the axial boundary by constructing equality constraints. The dimensionless expression is as follows: ； (7) In the formula, and This is the PINN approximation solution. and Given boundary values, and These are the initial predictions for the neural network.

[0012] Preferably, the loss function includes the Reynolds equation residual loss. Energy equation residual loss Pressure boundary loss and temperature boundary loss The specific expression is as follows: ; ; ; (8) In the formula, The number of configuration points calculated for PDE residuals. The number of configuration points calculated for boundary loss. , and Let be the dimensionless coordinates of the i-th configuration point.

[0013] Preferably, the gradient change index function in the hybrid configuration point generation strategy is as follows: (9) In the formula, p ( x ) is the gradient index probability density function of the PINN predicted solution. w ( x ) is the gradient index of the solution function. k It's a hyperparameter. To predict the gradient of the solution, the larger the value of k, the more concentrated the updated placement points are in the region with higher gradient.

[0014] Preferably, the dynamic loss weight adjustment strategy is based on Gaussian likelihood modeling, and the equivalent loss function is as follows: ; (10) In the formula, , , , The regularization term is used to measure the uncertainty of each loss term. The greater the uncertainty of a loss term, the smaller its corresponding weight.

[0015] Preferably, the loss function and parameter update method of the transfer learning strategy are as follows: Migration loss function: (11) In the formula, For PDE residual loss weights, For boundary loss weights, For migration weights, For variable parameters, For the target parameter, This represents the total loss of the PDE residuals. Total loss at the boundary; Parameter update formula: (12) In the formula, For learning rate, This represents the number of iterations.

[0016] Compared with the prior art, the beneficial effects of the present invention are: This invention boasts high computational efficiency, enabling rapid prediction of bearing thermal performance after training. It overcomes the time-consuming nature and difficulty in balancing accuracy and efficiency inherent in traditional numerical methods. It eliminates the need for large-scale high-fidelity simulation datasets, employing unsupervised model training by directly embedding physical equations into the loss function. This significantly reduces data preparation costs and addresses the dependence of data-driven models on datasets and the poor physical interpretability caused by the "black box" approach. It exhibits high accuracy and physical consistency, ensuring model convergence to an accurate solution that conforms to physical laws through hard constraint boundaries, dynamic loss weights, and dual optimizers. It possesses strong physical field coupling capabilities, accurately capturing the coupling relationship between the lubricating film temperature and pressure fields through a multi-network parallel architecture and parameter sharing, overcoming the limitation of existing PINN models that can only solve single physical fields. Furthermore, it demonstrates excellent generalization ability, rapidly extending the pre-trained model to different bearing structural parameters and operating conditions using transfer learning strategies. This meets the prediction needs of multi-condition and multi-structure designs, providing an efficient tool for bearing optimization design and real-time analysis. Attached Figure Description

[0017] Figure 1 This is a schematic diagram illustrating the coupling relationship between the Reynolds equation and the energy equation in the fluid radial sliding bearing thermodynamic performance prediction method based on physical information networks of this invention. Figure 2 This is a schematic diagram of the overall architecture of the PINN method for predicting the thermodynamic performance of fluid radial sliding bearings based on physical information networks, as proposed in this invention. Detailed Implementation

[0018] 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.

[0019] In bearing lubrication characteristic analysis methods, traditional numerical methods such as the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) rely on rigorous physical modeling and discretization strategies to achieve high computational accuracy under specific working conditions. However, their limitations are also quite prominent: First, the model construction is cumbersome, and the calculation accuracy is affected by the quality of the mesh generation; Second, the computational cost is high, and the solution time is huge when dealing with strongly nonlinear and multi-physics coupled problems; Third, its generalization ability is limited, making it difficult to meet the needs of real-time modeling and rapid prediction under multiple operating conditions.

[0020] In particular, when dealing with complex coupled problems such as viscosity-temperature effects of fluids (such as gas or oil film), gas compressibility, or oil film cavitation, traditional methods struggle to balance accuracy with solution efficiency.

[0021] To overcome the efficiency bottleneck of traditional numerical methods, intelligent algorithms, represented by machine learning, have opened up a new paradigm in recent years, especially reduced-order intelligent models (surrogate models) based on simulation data. These methods train neural networks to learn from large amounts of simulation data to achieve rapid performance prediction. However, such data-driven methods still have limitations: First, the accuracy of model predictions is highly dependent on the size and quality of the training dataset; Secondly, generating these large-scale, high-fidelity datasets is itself a computationally intensive task that requires time-consuming traditional numerical simulations to complete. Finally, the "black box" nature of the model makes its physical interpretability poor, and the prediction results may lack physical consistency.

[0022] Physics-Informed Neural Networks (PINN) offer a novel approach to solving these challenges. PINN directly embeds physical conservation laws (such as partial differential equations) into the loss function of the neural network, guiding model training without requiring large amounts of labeled data, thus solving complex physical problems in an unsupervised or semi-supervised manner. Inspired by this, PINN has been initially applied in bearing lubrication, for example, to solve the (generalized) Reynolds equation, achieving good results in pressure field distribution calculations or cavitation boundary identification. However, current applications of PINN in bearing analysis are mainly limited to solving the (generalized) Reynolds equation in a single physics field. In the multiphysics thin-film lubrication problem involving the variable viscosity generalized Reynolds equation and its coupling with the energy equation, PINN still holds great potential and warrants further exploration and development.

[0023] To fill the gaps in the aforementioned research, this invention proposes a method for predicting the thermodynamic performance of fluid radial sliding bearings based on physical information networks (PINNs). This method constructs a parallel multi-network architecture, explicitly incorporating the strong coupling physical mechanism of fluid compressibility (for gas bearings) or cavitation effect (for oil film bearings) with the viscosity-temperature effect into the PINN loss function. It also combines hard constraint boundaries, dynamic loss weights, hybrid optimizers, and transfer learning strategies to overcome the limitations of high computational cost in traditional methods and the dataset dependence of data-driven methods, achieving fast, high-precision, and highly generalizable prediction of bearing multi-physics fields.

[0024] Please see Figure 1-2 This invention provides a technical solution: a method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network, comprising the following steps: S1. Construct a core mathematical model characterizing the thermo-mechanical coupling characteristics of a radial sliding bearing. This core model includes the generalized two-dimensional, variable-viscosity, compressible, and steady-state Reynolds equation, two-dimensional steady-state energy equation, lubricating film thickness equation, and velocity field equation to characterize the pressure distribution of the lubricating film in the radial sliding bearing. Dimensionless processing is applied to each equation to eliminate training interference caused by differences in the magnitude of physical quantities. The thermo-mechanical coupling characteristics of the radial sliding bearing are mainly related to the Reynolds equation and the energy equation, considering fluid compressibility (for gas bearings) or cavitation effects (for oil film bearings) and viscosity-temperature effects. The Reynolds equation is: (1) in, For the pressure distribution of the lubricating film, These are the circumferential, film thickness, and axial coordinates, respectively. It refers to the thickness of the lubricating film. It refers to the viscosity of the lubricating film, which is related to the temperature of the lubricating film. It is the journal speed. It is the journal rotational angular velocity. It is the bearing radius; The expression for the lubricating film thickness equation is: (2); in It is the radial clearance of the bearing. It is the bearing eccentricity; The two-dimensional steady-state energy equation, neglecting heat transfer and convection changes in temperature along the film thickness direction, is expressed as: (4) in, It is the two-dimensional temperature field distribution of the lubricating film. It is air density. It is specific heat capacity. It is the thermal conductivity, which is used to obtain the pressure field distribution of the lubricating film. Then, the velocity field of the lubricating film can be solved using the gradient of the pressure field. This velocity field is the superposition of the shear flow and the pressure gradient-driven flow. The velocity field equation includes the circumferential and axial velocity components, respectively. and The expression is obtained by solving for the pressure field gradient of the lubricating film, as shown below: ； (3); Since the order of magnitude of lubricating film pressure is usually in... arrive Between, and the order of magnitude of temperature is approximately The dimensions of lubricating film thickness, bearing width, and diameter are on the order of magnitude smaller, typically within... The magnitude is on the order of magnitude. Therefore, in order to avoid problems such as gradient vanishing, gradient explosion, and unbalanced weight updates during PINN training caused by excessive differences in magnitude, this paper performs dimensionless processing on equations (1)-(4). Dimensionless processing is a normalization transformation of physical quantities such as lubricating film pressure, temperature, thickness, and bearing size in the coupled mathematical model that have large differences in magnitude, so that the dimensionless values ​​of each physical quantity are in the same order of magnitude range. The dimensionless equations after processing are: (5) In the formula, , and The coordinates are dimensionless. , , , and The pressure, temperature, velocity components, and lubricating film thickness are dimensionless. , - These are dimensionless coefficients. This refers to the axial length of the bearing.

[0025] The coupling mechanism between the Reynolds equation and the energy equation lies in the real-time updates of the lubricating film viscosity, density, and velocity field. The temperature field, as a solution to the energy equation, directly and significantly alters the lubricating film properties required to solve the Reynolds equation; for example, viscosity is a function of temperature. The changing viscosity field is substituted into the Reynolds equation as a variable coefficient, thus directly affecting the intensity of the hydrodynamic effect and pressure distribution. For compressible lubricating media, the lubricating film density is updated through the ideal gas law, influenced not only by the pressure field solved by the Reynolds equation but also by the temperature field solved by the energy equation, making it a key variable connecting the two equations. Correspondingly, the pressure field determined by the Reynolds equation drives the velocity field within the gas film. , The velocity field determines, on the one hand, the magnitude of the convection term in the energy equation, i.e., the rate at which heat is transported; on the other hand, it also determines the magnitude of the viscous dissipation term, i.e., the rate at which frictional heat is generated. The coupling effects between these multiple physical quantities constitute the core physical mechanism of bearing thermal-fluid coupling analysis, as shown in the appendix. Figure 1 As shown.

[0026] S2. Considering the coupling relationship between the temperature and pressure fields of the bearing lubrication film, a multi-network parallel Physical Information Neural Network (PINN) architecture is designed. This architecture contains at least two fully connected neural networks, one for predicting the lubrication film pressure and the other for predicting the lubrication film temperature. Parameter sharing between the two independent networks effectively captures the coupling relationship between the Reynolds equation and the energy equation for radial sliding bearings, thus enabling accurate prediction of the pressure and temperature fields of the bearing lubrication film. The values ​​of the predicted variables and partial derivatives are passed between the networks to fit the coupling relationship between the Reynolds equation and the energy equation. The fully connected neural network consists of the following: inputs are the coordinates of data points (x, z) in the computational domain; 8 hidden layers, each with 64 neurons; and outputs are pressure and temperature fields. The network first uses two independent linear layers to compute two fixed-value gated encoder vectors from the input. Subsequently, in each hidden layer of the network backbone, the activation values ​​of that layer are used as a dynamic gate to weight and mix the two fixed encoder vectors. This "adjusted" feature vector is then used as the input to the next layer. This self-attention mechanism allows the information from the original input to continuously influence the feature representations of deeper layers of the network in a non-linear, layer-by-layer adjusting manner.

[0027] Specifically, the lubricating film temperature obtained by the temperature prediction network is used to update the viscosity and substituted into the Reynolds equation to recalculate its PDE residuals; while the lubricating film pressure obtained by the pressure prediction network can be used to solve the lubricating film velocity field, and then combined with the ideal gas law to update the density, which is used to correct the convection and dissipation terms in the energy equation. Note that all networks are trained in the same computational domain and have the same network structure, the overall structure of which is shown in the attached figure. Figure 2 As shown. In order to better capture the complex nonlinear coupling relationship between the Reynolds equation and the energy equation, this paper introduces a self-attention mechanism on the basis of the multilayer perceptron. By providing diverse attention weights through two sets of encoders, the features of each layer are dynamically weighted and combined to adapt to the complex coupling relationship between multiphysics fields.

[0028] PINN's neural network consists of multiple fully connected layers, each employing the Tanh activation function for nonlinear transformation. The Tanh function is chosen to provide a smooth nonlinear mapping, enabling the network to effectively capture the nonlinear characteristics of complex physical scenarios, especially when dealing with coupled multi-physics problems. The Tanh activation function helps mitigate the vanishing gradient problem, allowing for efficient training of deep networks and thus improving PINN's performance in multi-physics coupled problems. To enhance training stability and convergence speed, the network's weights are initialized using a Xavier normal distribution. This initialization strategy avoids the vanishing and exploding gradient problems by balancing the variance of the input and output layer weights, ensuring efficient gradient propagation during PINN training.

[0029] S3. Construct a composite loss function consisting of the residual terms of the partial differential equation (PDE) and boundary loss terms. Use a hard constraint method to force the axial boundary conditions to converge to a given environmental value. Apply periodic soft constraints to the circumferential boundary to ensure the continuous smoothness of the physical quantities and their first derivatives. The axial boundary conditions include axial end-face boundaries and circumferential periodic boundaries. For the entire lubrication domain of the bearing, the complete boundary conditions include both axial end-face boundaries and circumferential periodic boundaries. At the two axial end faces of the bearing, the lubricating film is in communication with the external environment. Therefore, it is assumed that its pressure is equal to the environmental pressure. For temperature, it is usually also given as ambient temperature. Furthermore, due to the circular geometry of the bearing, its physical field must be continuous and smooth in the circumferential direction. Therefore, at the initial position in the circumferential direction... and termination position Periodic boundary conditions are applied between the physical quantities. These conditions require not only that the numerical values ​​of the physical quantities be equal, but also that their first derivatives (gradients) be equal, to ensure the continuity and smoothness of the solution, as detailed below: Axial end face boundary: The dimensionless lubrication film pressure satisfies Temperature meets ; Circumferential periodic boundary: circumferential starting position after dimensionless transformation and termination position At this point, the physical quantity's numerical value and its first derivative are all equal, as expressed below: ; (6) The PINN training process solves an approximate PDE solution by minimizing the PDE residuals and matching the boundary prediction results to the target data. However, when dealing with highly nonlinear PDEs, this method may lead to fluctuations or even inaccuracies in the boundary prediction results. This is mainly due to the optimization competition between the soft constraint loss term and the PDE residual term in the boundary conditions. This invention is based on a hard constraint method, which constructs equality constraints to force the Reynolds equation and energy equation to converge to a given boundary value at the axial boundary; while for circumferential periodic boundaries, a soft constraint method is still used. This strategy effectively avoids the problem of difficult convergence of the boundary loss and accelerates the convergence speed of the loss function during the overall training process. Specifically: The hard constraint method forces the Reynolds equation and the energy equation to converge to a given boundary value at the axial boundary by constructing equality constraints. The dimensionless expression is as follows: ； (7) In the formula, and This is the PINN approximation solution. and Given boundary values, and These are the initial predictions for the neural network; Combining equations (5)-(7), the loss function used to evaluate the convergence of PINN includes the Reynolds equation residual loss. Energy equation residual loss Pressure boundary loss and temperature boundary loss The specific expression is as follows: ; ; ; (8) In the formula, The number of configuration points calculated for PDE residuals. The number of configuration points calculated for boundary loss. , and Let be the dimensionless coordinates of the i-th configuration point.

[0030] S4. The placement points of PINN are derived from random point sampling of the physical computational domain, similar to mesh generation in finite element methods. In early PINN training, placement points were typically static placement points determined by uniform random sampling before training. For some simple PDEs, the neural network converged relatively easily. However, in complex physical scenarios where the pressure and temperature fields of the bearing lubrication film are coupled, the non-uniformity of the placement points makes PINN prone to failing to converge to an accurate solution in critical regions or regions with large gradient changes. This invention adopts a hybrid placement point method. Before PINN training, fixed static placement points are generated in the computational domain through Latin hypercube sampling. During training, the placement points are adaptively updated based on the gradient change index, and together with the pre-set fixed placement points, they constitute the PINN training input. The gradient index function used to determine the placement point update is as follows: (9) In the formula, p ( x ) is the gradient index probability density function of the PINN predicted solution. w ( x ) is the gradient index of the solution function. k It's a hyperparameter. To predict the gradient of the solution, when k When =0, p ( x The distribution is uniform; the larger the value of k, the more concentrated the updated configuration points are in the region with higher gradient. The loss functions of S5 and PINN consist of multiple loss terms, and their training process can be viewed as a multi-objective optimization problem. During training, these heterogeneous loss terms often exhibit significant differences in magnitude and gradient competition, making the training process using a fixed-weight strategy prone to getting stuck in local optima or stagnating. Especially in the complex multi-physics coupling scenario of hydrodynamic sliding bearings, this rigid weight allocation scheme further exacerbates the imbalance in the optimization process. Meanwhile, tedious manual parameter tuning is not only time-consuming and inefficient but also fails to guarantee optimal results. This invention designs a dynamic loss weight allocation mechanism based on uncertainty measurement, performs Gaussian likelihood modeling on each loss term, automatically quantifies model confidence, and dynamically adjusts the weights to avoid training getting stuck in local optima or stagnating. The dynamic loss weight adjustment strategy is based on Gaussian likelihood modeling, and the equivalent loss function is as follows: ; (10) In the formula, , , , The regularization term is used to measure the uncertainty of each loss term; the greater the uncertainty of a loss term, the smaller its corresponding weight. S6. To achieve a balance between efficiency and accuracy in PINN training, a hybrid optimizer and adaptive annealing learning rate strategy are used to train the PINN model. In the early stages of training, the Adam optimizer is used for global search to achieve rapid convergence and reduce the risk of getting trapped in local optima. As the model parameters gradually approach the optimal solution, the L-BFGS optimizer is switched to perform fine-tuning optimization, thereby improving prediction accuracy. In addition, an adaptive annealing learning rate strategy is introduced during training to dynamically adjust the learning rate, preventing the model from getting trapped in suboptimal solutions due to excessively rapid convergence, and enhancing training stability. By gradually reducing the learning rate, this method effectively improves the accuracy and physical rationality of the solution while ensuring training efficiency, thereby enhancing PINN's performance in solving complex physically coupled problems and avoiding trivial solutions that formally satisfy PDEs but lack physical meaning. The training process of PINN is shown in Table 1. Table 1: PINN Training Process S7. The input of the above network structure is only discrete data points within the computational domain, and it is only applicable to the prediction of bearing lubrication characteristics under fixed structures and working conditions. In fact, in engineering applications, bearings often face complex working environments, such as variable working conditions and multi-structure designs, requiring continuous optimization of various parameters during the bearing design process to achieve optimal performance in different application scenarios. Therefore, the proposed PINN order reduction alternative model needs to have higher generalization ability so that it can still achieve accurate prediction of multi-physics fields when facing bearings with different structures. To address this issue, this patent adopts a PINN transfer learning design scheme based on the minimum energy path, which initializes the required extended PDE parameters through prior PINN intelligent initialization and treats them as trainable weights. During the training process, these prior parameters will be adaptively updated as the model is trained to achieve effective extension and optimization of PINN in multi-physics and multi-structure environments. Based on Equation (10), the designed loss term is as follows: Migration loss function: (11) In the formula, For PDE residual loss weights, For boundary loss weights, For migration weights, For variable parameters, For the target parameter, This represents the total loss of the PDE residuals. Total loss at the boundary; Parameter update formula: (12) In the formula, For learning rate, The number of iterations represents the weights of the neural network during each update. It will be based on the parameters The changes are adjusted so that PINN can gradually approach the solution of the target task. The specific process of the migration strategy is shown in Table 2. Table 2: Migration Strategies To address the issues of traditional numerical methods (FDM / FEM / FVM) being cumbersome in model construction, computationally expensive, and having limited generalization ability, data-driven models relying on large-scale high-fidelity simulation datasets and having poor physical interpretability, and existing PINN methods being limited to solving single physics fields, this invention constructs a multi-network parallel PINN architecture. It embeds the residuals of the coupled Reynolds equation and energy equation into the loss function, combines hard constraint methods to force axial boundary conditions to converge to given boundary values, hybrid configuration point generation (Latin hypercube sampling static points and gradient-adaptive dynamic points), dynamic loss weight adjustment based on uncertainty metrics, and a training strategy using dual optimizers (Adam global search and L-BFGS fine-tuning) and adaptive annealing learning rate. Simultaneously, it introduces a transfer learning strategy (pre-trained parameter initialization and target parameter gradient descent update). This invention enables unsupervised learning without the need for large-scale simulation data. After training, single predictions are completed in milliseconds (e.g., 0.0333 seconds, more than 300 times faster). It possesses high precision, physical consistency, and strong generalization ability. It can accurately capture the multi-physics coupling relationship between the pressure field and temperature field of the bearing lubrication film. It is adaptable to different bearing structural parameters (e.g., radial clearance, eccentricity) and operating conditions (e.g., speed). It is applicable to compressible gas bearings and incompressible oil film bearings. Its key points and protection points include multi-network parallel coupling architecture, physical equation residual embedding loss function, hard constraint boundary method, dynamic loss weight adjustment, hybrid optimizer strategy, transfer learning steps, etc. Traditional numerical calculation, data-driven order reduction prediction, parameter extension PINN and other alternative solutions all have problems with efficiency, cost or insufficient generalization ability and cannot replace this patented solution.

[0031] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0032] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network, characterized in that: Includes the following steps: S1. Construct a mathematical model characterizing the thermo-mechanical coupling characteristics of a fluid radial sliding bearing. The mathematical model includes the Reynolds equation, the two-dimensional steady-state energy equation, the lubricating film thickness equation, and the velocity field equation, and performs dimensionless processing on each equation. S2. Design a multi-network parallel physical information network (PINN) architecture, including at least two fully connected neural networks, which are used to predict the pressure field and temperature field of the lubricating film, respectively. The physical equations are coupled between the networks through parameter sharing. S3. Construct a composite loss function that includes the residuals of partial differential equations (PDEs), boundary losses, and parameter penalty terms. Hard constraints are used to force the axial boundary conditions to converge to the given boundary values, while soft constraints are used to handle the circumferential periodic boundaries. S4. A hybrid placement point generation strategy is adopted, which includes: generating fixed static placement points through Latin hypercube sampling; adaptively updating dynamic placement points based on gradient change indicators during training; and using both static and dynamic placement points as training inputs. S5. Design a dynamic loss weight adjustment strategy based on uncertainty measurement to automatically quantify the confidence level of each loss item and assign weights. S6. A training strategy using dual optimizers and adaptive annealing learning rate is adopted. First, the Adam optimizer is used for global search, and then the L-BFGS optimizer is switched to perform fine optimization. S7. Combining transfer learning strategies, the pre-trained model is extended to new bearing structural parameters or operating conditions through gradient descent, enabling rapid prediction of thermal performance.

2. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The Reynolds equation is used to characterize the generalized two-dimensional, variable-viscosity, compressible, and steady-state pressure distribution of the lubricating film in radial sliding bearings. Considering fluid compressibility or cavitation effects and viscosity-temperature effects, its equation is as follows: （1） in, For the pressure distribution of the lubricating film, These are the circumferential, film thickness, and axial coordinates, respectively. It refers to the thickness of the lubricating film. It refers to the viscosity of the lubricating film, which is related to the temperature of the lubricating film. It is the journal speed. It is the journal rotational angular velocity. It is the bearing radius; The expression for the lubricating film thickness equation is as follows: （2） in It is the radial clearance of the bearing. It is the bearing eccentricity; The two-dimensional steady-state energy equation neglects heat transfer and convection changes in temperature along the film thickness direction, and its expression is: （4） in, It is the two-dimensional temperature field distribution of the lubricating film. It is air density. It is specific heat capacity. It is the thermal conductivity, and the velocity field equations include the circumferential and axial velocity components, respectively. and The expression is obtained by solving for the pressure field gradient of the lubricating film, as shown below: ； （3）。 3. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The dimensionless transformation process involves normalizing physical quantities in the coupled mathematical model, such as lubricating film pressure, temperature, thickness, and bearing size, which have significant differences in magnitude. This normalization ensures that the dimensionless values ​​of each physical quantity are within the same order of magnitude. The resulting dimensionless equation is: （5） In the formula, , and The coordinates are dimensionless. , , , and The pressure, temperature, velocity components, and lubricating film thickness are dimensionless. , - These are dimensionless coefficients. This refers to the axial length of the bearing.

4. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The multi-network parallel Physical Information Neural Network (PINN) architecture also satisfies: The temperature field output by the temperature prediction network is used to update the viscosity of the lubricating film, and then substituted into the Reynolds equation to calculate the residuals of the partial differential equation (PDE). The pressure field output by the pressure prediction network is used to solve the velocity field, and the density is updated in combination with the ideal gas law to correct the convection and dissipation terms in the energy equation. The network is based on a multilayer perceptron and introduces a self-attention mechanism. It generates diverse attention weights through two sets of encoders and dynamically weights and combines the features of each layer. The Tanh activation function is used to achieve smooth nonlinear mapping and alleviate the gradient vanishing problem; The weights are initialized using the Xavier normal distribution to balance the variance of the input and output layers and ensure effective gradient propagation.

5. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The axial boundary conditions include axial end face boundaries and circumferential periodic boundaries, as detailed below: Axial end face boundary: The dimensionless lubrication film pressure satisfies Temperature meets ; Circumferential periodic boundary: circumferential starting position after dimensionless transformation and termination position At this point, the physical quantity's numerical value and its first derivative are all equal, as expressed below: ； （6）。 6. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The hard constraint method forces the Reynolds equation and the energy equation to converge to a given boundary value at the axial boundary by constructing equality constraints. The dimensionless expression is as follows: ； （7） In the formula, and This is the PINN approximation solution. and Given boundary values, and These are the initial predictions for the neural network.

7. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The loss function includes the Reynolds equation residual loss. Energy equation residual loss Pressure boundary loss and temperature boundary loss The specific expression is as follows: ； ； ； （8） In the formula, The number of configuration points calculated for PDE residuals. The number of configuration points calculated for boundary loss. , and Let be the dimensionless coordinates of the i-th configuration point.

8. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The gradient change index function in the hybrid configuration point generation strategy is as follows: （9） In the formula, p ( x ) is the gradient index probability density function of the PINN predicted solution. w ( x ) is the gradient index of the solution function. k It's a hyperparameter. To predict the gradient of the solution, the larger the value of k, the more concentrated the updated placement points are in the region with higher gradient.

9. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The dynamic loss weight adjustment strategy is based on Gaussian likelihood modeling, and the equivalent loss function is as follows: ； （10） In the formula, , , , The regularization term is used to measure the uncertainty of each loss term. The greater the uncertainty of a loss term, the smaller its corresponding weight.

10. The method for predicting the thermodynamic performance of a fluid radial sliding bearing based on a physical information network according to claim 1, characterized in that: The loss function and parameter update method of the transfer learning strategy are as follows: Migration loss function: (（11） In the formula, For PDE residual loss weights, For boundary loss weights, For migration weights, For variable parameters, For the target parameter, This represents the total loss of the PDE residuals. Total loss at the boundary; Parameter update formula: （12） In the formula, For learning rate, This represents the number of iterations.

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