A static pressure torus throttling gas bearing load capacity prediction method based on GA-BP neural network

Abstract

This invention provides a method for predicting the load-carrying capacity of hydrostatic toroidal throttling gas bearings based on a GA-BP neural network, belonging to the field of hydrostatic gas bearing performance prediction. Compared with the traditional GA-BP approach, which typically uses training error as the sole optimization objective and involves long training rounds for each individual, resulting in high computational cost and a tendency to overfit, this invention introduces short-round BP training during the genetic search phase to quickly evaluate the merits of candidate weight biases. This significantly reduces the computational cost of fitness evaluation and accelerates convergence. Simultaneously, a hybrid fitness function incorporating training set error and validation set error is constructed, allowing genetic evolution to simultaneously constrain generalization bias while improving fitting accuracy. This effectively overcomes the problem of traditional GA-BP, which only pursues optimal training set results and suffers from unstable generalization. After obtaining the initial weights and biases corresponding to the chromosome with optimal fitness, joint training-validation set constraint retraining is performed on the sample data to further refine the network parameters and improve prediction stability and accuracy.

Description

Technical Field

[0001] This invention relates to the field of performance prediction of hydrostatic gas bearings, and more particularly to a method for predicting the load-bearing capacity of hydrostatic toroidal throttling gas bearings based on a GA-BP neural network. Background Technology

[0002] Gas bearings are generally classified into hydrostatic gas bearings and dynamic gas bearings based on the formation of gas film pressure. Dynamic gas bearings rely on the relative motion between the rotor and the bearing surface to generate dynamic pressure in the gas film, forming the load-bearing capacity. Their load-bearing capacity is related to the rotational speed; the gas film is difficult to establish during low-speed start-up and shutdown, easily leading to rubbing and fluctuations in accuracy. Hydrostatic gas bearings, on the other hand, continuously supply compressed gas to the bearing gap through a gas source device, forming a stable gas film even at zero or low speeds. They offer advantages such as low friction, high precision, minimal thermal deformation, and no pollution, making them more suitable for applications requiring extremely high micro-vibration and motion precision. Hydrostatic gas bearings are widely used in precision milling machines, ultra-precision machine tool spindles, and air-bearing guides, and are further extended to high-precision positioning platforms and precision measurement and calibration equipment. Their load-bearing capacity directly determines the system's operational stability, accuracy, and service life; therefore, accurately and quickly predicting the gas bearing's load-bearing capacity is a crucial prerequisite for optimizing the gas bearing structure design.

[0003] The current methods for assessing and predicting the load-bearing capacity of hydrostatic gas bearings typically employ three approaches: First, obtaining load-bearing capacity data through testing on a test bench. However, this method is limited by testing costs and time constraints, making it difficult to rapidly and systematically cover multi-factor combined operating conditions. Second, numerical solutions and iterative calculations based on fluid control equations. This method can obtain high-precision results, but the solution process is complex and computationally time-consuming, making it difficult to achieve rapid response during the structural and parameter optimization stages. Third, estimations using empirical formulas are commonly used in engineering design. However, these are often based on specific structures and operating conditions, with clearly defined applicability boundaries. They are prone to deviations when faced with changes in throttling structures or cross-range parameter adjustments, making it difficult to balance universality and prediction accuracy. Therefore, existing methods struggle to simultaneously meet the demand for high-precision prediction under complex multi-parameter operating conditions.

[0004] The load-bearing capacity of hydrostatic gas bearings exhibits a strong nonlinear coupling relationship with structural and operating parameters, making it difficult for traditional methods to balance versatility and accuracy. In existing technologies, backpropagation (BP) neural networks can be used to fit nonlinear mappings, but their training is easily affected by initial parameters and learning rate settings, leading to convergence instability and the risk of local optima. To address this, existing techniques have proposed using genetic algorithms to optimize the weights and biases of BP networks, improving initial value sensitivity and enhancing prediction performance. However, existing methods still have room for improvement in terms of prediction accuracy, computational efficiency, and applicability. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a method for predicting the bearing capacity of hydrostatic toroidal throttling gas bearings based on a GA-BP neural network. Compared to traditional GA-BP methods, which typically use training error as the sole optimization objective and involve long training rounds for each individual, leading to high computational costs and a tendency to overfit, this invention introduces short-round BP training during the genetic search phase to quickly evaluate the merits of candidate weight biases. This significantly reduces the computational cost of fitness evaluation and accelerates convergence. Simultaneously, a hybrid fitness function incorporating both training and validation set errors is constructed, allowing genetic evolution to simultaneously constrain generalization bias while improving fitting accuracy. This effectively overcomes the problem of traditional GA-BP methods that only pursue optimal training set results in unstable generalization. After obtaining the initial weights and biases corresponding to the chromosome with optimal fitness, joint training-validation set constraint retraining is performed on the sample data to further refine the network parameters, improve prediction stability and consistency, and provide a basis for optimizing structural parameters.

[0006] The technical means employed in this invention are as follows:

[0007] A method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network includes: S1. Using a design sampling method, gas bearing characteristic data is generated within the preset range of structural and operating parameters. For each set of parameter combinations in the gas bearing characteristic data, the corresponding load-bearing capacity data is obtained through CFD numerical simulation or experimental testing. Multiple sets of sample datasets are constructed, and the sample datasets are divided into training set, validation set and test set according to a preset ratio. S2. Determine the network structure parameters, and use the gas bearing characteristic data as the network input and the bearing capacity data as the output to normalize the sample dataset. S3. Construct a BP neural network model, using normalized gas bearing feature data as input and normalized bearing capacity data as output. Train the network with different numbers of hidden layer nodes on the training set, and use the minimum mean square error of the validation set as the selection criterion for the number of hidden layer nodes. Train the constructed BP neural network model to determine the optimal number of hidden layer nodes, so as to ensure that the BP neural network model has the best fitting ability to the gas bearing bearing capacity data. S4. The weights and bias parameters of the BP neural network are used to form a set of parameters to be optimized, and the set of parameters to be optimized is encoded with real numbers to generate chromosome individuals for the genetic algorithm. S5. After determining the BP neural network structure, construct a GA-BP network model by combining the genetic algorithm; concatenate and encode the weight parameter segment and bias parameter segment of the network model into a chromosome individual in a preset order, and determine the value range of each parameter segment according to the Xavier / Glorot initialization rule, and set symmetrical upper and lower boundaries for each gene in the chromosome to constrain the chromosome search space. S6. Based on the generated chromosomes, construct a hybrid fitness function for the GA-BP network model to balance fitting accuracy and generalization ability. S7. Genetic operations are performed based on the hybrid fitness function. The elite retention strategy is used to retain the current best individual. The population is updated iteratively to obtain the chromosome with the best fitness. The optimal weight matrix and optimal bias vector of the corresponding GA-BP network model are then decoded. S8. Based on the weights and biases obtained from decoding, perform joint constraint retraining of the GA-BP network model using the training and validation sets to further optimize the weights and biases of the GA-BP network model. S9. Based on the further optimized weights and biases, forward prediction and inverse normalization are performed to finally obtain the predicted value of the hydrostatic gas bearing capacity.

[0008] Furthermore, in step S1, the constructed sample dataset includes gas bearing feature data and load-bearing capacity data, wherein: The gas bearing characteristic data includes structural parameters and operating parameters. The structural parameters include orifice diameter, number of orifice rows, spacing between rows, number of orifices, end-to-end distance ratio, length-to-diameter ratio, orifice length, design gas film thickness, and bearing inner diameter. The operating parameters include supply pressure, rotational speed, and steady-state load. The gas bearing characteristic data can be expressed as follows:

[0009] in, Represents the characteristic data matrix of gas bearings. This indicates the total number of gas bearing samples. Representing feature dimension, Indicates the first Group gas bearing characteristic data; The bearing capacity data is represented as follows:

[0010] in, Represents a bearing capacity data vector. Indicates and The corresponding load-bearing capacity data.

[0011] Further, step S2 includes: S21. Let the number of nodes in the input layer be... , Consistent with the characteristic dimensions of gas bearings, let the number of output layer nodes be... The corresponding bearing capacity value; S22, Gas bearing feature data matrix Defined as input matrix , load-bearing capacity data vector Defined as output vector ; S23, Definition of the 3D features, as follows:

[0012] in, Indicates the first The sample at the th Values ​​on the dimensional features; Indicates the first in the training set The minimum value of a feature; Indicates the first in the training set The maximum value of a feature; S24, Based on the definition of the first The gas bearing feature data, including structural parameters and operating parameters, is used as training input features and linearly normalized. Each feature dimension is mapped to the [0,1] interval, and the normalization transformation yields:

[0013] in, express The normalized value; S25, Using the training set The minimum and maximum values ​​of the dimensional features are used to perform the same transformation on the gas bearing feature data of the validation and test sets to ensure consistent data scale. S26. Using the bearing capacity data as the training output, perform linear normalization, mapping each feature dimension to the [0,1] interval, and then normalizing to obtain:

[0014] in, Indicates the first The true label value of the nth sample, i.e., the nth Load-bearing capacity data corresponding to the group of gas bearing samples; This represents the minimum value of the training set labels; This represents the maximum value of the training set labels.

[0015] Further, step S3 includes: S31. Let the search range for the number of hidden layer nodes be:

[0016] in, Indicates the number of hidden layer nodes; S32, For each hidden layer node number Construct a single-hidden-layer feedforward backpropagation neural network model, including an input layer, a hidden layer, and an output layer, where the input layer contains... The hidden layer contains 10 neurons for receiving input features; 10 neurons; the output layer contains 100 neurons; Each neuron outputs a prediction result; S33. Calculate the mean square error of the training set, using the following formula:

[0017] in, Indicates in The mean squared error of the training set with a number of hidden layer nodes; Indicates the number of samples in the training set; A set of indices representing the samples in the training set; Indicates the normalized i-th The true values ​​of the training samples in the training set; Represents the normalized first i Predicted values ​​for each training sample in the training set; S34. Use a BP neural network model to test the normalized validation set. Make predictions and obtain the results. Validation set prediction output with a number of hidden layer nodes ; S35. Calculate the mean square error of the verification set, using the following formula:

[0018] in, Indicates in The mean squared error of the validation set with a number of hidden layer nodes; Indicates the number of samples in the validation set; A set of indices representing the validation set samples; Indicates the normalized i-th The true values ​​of the training samples in the validation set; Indicates the normalized i-th Predicted values ​​of training samples in the validation set; S36. Select a value that makes the mean square error of the validation set... The minimum number of hidden layer nodes is found, and the optimal number of hidden layer nodes is as follows:

[0019] in, This represents the optimal number of nodes in the hidden layer.

[0020] Further, step S4 includes: S41. Suppose that the trained BP neural network model contains a weight matrix from the input layer to the hidden layer. ∈ Bias vector of hidden layer ∈ Weight matrix from hidden layer to output layer ∈ and the bias vector of the output layer ∈ ; S42. Expand the weights and biases from step S41 into a real chromosome vector in a fixed order, as follows:

[0021] in, Indicates chromosome length, ; Further, step S5 includes: S51. Set the weight boundaries from the input layer to the hidden layer as follows: , ∈[- , ] S52. Set the offset boundary of the hidden layer as follows: , ∈[- , ] S53. Set the weight boundaries from the hidden layer to the output layer as follows: , ∈[- , ] S54. Set the offset boundary of the output layer as follows: , ∈[- , ] S55. The upper and lower boundaries of each gene are determined by the parameter boundary set. , , It is assembled by segmenting according to the sequence of chromosome gene combinations, and each gene corresponds to , , , One of the scalar parameter elements; S56. Set the genetic algorithm parameters, including the maximum number of generations, population size, crossover probability, and mutation probability; S57, For each individual k =1, ..., sizepop, generate chromosomes by randomly sampling within the corresponding boundaries: ,

[0022] in, Indicates the first Individual chromosome vector The Middle The values ​​of each gene; Indicates the first The lower boundary of a gene; Indicates the first The upper boundary of a gene; This represents the random number used for random sampling.

[0023] Further, step S6 includes: S61, regarding the first k Decoding each chromosome yields... , , and And assign these values ​​to the GA-BP network model as initial parameters of the GA-BP network model; S62, Training Set Perform short-round backpropagation training to obtain the predicted output set. ; S63. Perform forward propagation on the validation set to obtain the validation set prediction output set. ; S64, Calculate the first The mean absolute error of the training set during BP training for each individual short-wheel number reflects the fitting accuracy of the GA-BP network model to the known gas bearing load capacity data. The formula is as follows: ,

[0024] in, No. Each individual represents the mean absolute error of the training set during short-round backpropagation training; This indicates the number of short-round backpropagation (BP) training cycles. Normalized predicted values ​​of training samples in the individual training set; S65, Calculate the first The mean absolute error of the validation set during training of individual short-wheel-count backpropagation (BP) models reflects the model's ability to generalize to unseen gas bearing capacity data. The formula is as follows: ,

[0025] in, Indicates the first The mean absolute error of the validation set during short-round backpropagation training of each individual Indicates the first round after short-round training Normalized predicted values ​​of training samples in the individual validation set; S66. Calculate the error of each individual in the population after short-round backpropagation training independently as the mixed fitness function:

[0026] in, Represents the mixed fitness function. The weight parameters for the validation set.

[0027] Further, step S7 includes: S71. Select high-quality individuals based on the hybrid fitness function, and calculate the... The selection weights for each individual are calculated using the following formula:

[0028] in, Indicates the first The selection weights corresponding to each individual; This represents a constant to prevent the denominator from being zero; S72, the first calculation-based The selection weight corresponding to each individual The probability of an individual's choice is calculated using the following formula:

[0029] S73, Using roulette wheel betting method Sizepop individuals are extracted from the original population to form a new generation population, retaining high-quality gas bearing parameters, namely the network weights and biases corresponding to the gas bearing characteristic data.

[0030] S74. Crossover of the parent individuals obtained through selection operations to generate offspring, achieving chromosomal gene recombination, specifically including: For the two parent individuals to be crossed and Randomly select a gene location ,make:

[0031]

[0032]

[0033] And Within the corresponding boundary range, high-quality genes are integrated; among them, Indicates the parent generation In the The values ​​of each gene locus; Indicates the parent generation In the The values ​​of each gene locus; The crossover coefficient is represented and follows a uniform distribution. These represent the two offspring in the [number]th generation. The values ​​of each gene locus; S75. Perform mutation operations on offspring individuals according to the mutation probability, specifically including: For the probability of mutation Selected individuals and gene locations Define the step size factor that varies with the number of generations:

[0034] in, Indicates the variable-asynchronous length coefficient; Represents a random number that follows a uniform distribution; Indicates the current generation number; Indicates the maximum number of generations; If random number Then, it mutates upwards, resulting in... ( ) Otherwise, it mutates downwards, resulting in... ( ) and will Crop to boundary [ , Within this range, increasing population diversity helps avoid getting trapped in local optima; in the above formula, Indicates the chromosome before the mutation The One gene; This represents the new gene value obtained after the mutation; Indicates the first The lower boundary of a gene; Indicates the first The upper boundary of a gene; S76. Design elite retention strategy, as follows: At the end of each generation, compare the error of the best individual in the current generation. Error compared to historical best ,like If the optimal individual is updated, the individual with the largest error in the current population is replaced with the optimal individual to ensure that the network weights and biases with the smallest prediction error of the gas bearing capacity of the GA-BP network model are not lost after the genetic algorithm is optimized. S77, Based on maximum number of evolutions Repeat steps S71-S76 to obtain the chromosome with optimal fitness. ; S78, after completion After generations of evolution, based on the chromosome with the best fitness Decoding yields the optimal initial weight matrix and bias vector for the corresponding BP network: , , ,

[0035] in, This represents the optimal weight matrix from the input layer to the hidden layer; This represents the optimal bias vector of the hidden layer; This represents the optimal weight matrix from the hidden layer to the output layer; This represents the optimal bias vector of the output layer.

[0036] Further, step S8 includes: S81. Concatenate the normalized training set, validation set, and test set: = , =

[0037] in, This represents the fully normalized gas bearing characteristic data. The test set is used only for the final forward prediction and performance evaluation, and does not participate in error backpropagation and parameter updates during the training process, as it contains fully normalized bearing capacity data. S82, Set the training set, validation set, and test set respectively. Index range in:

[0038]

[0039]

[0040] S83. Based on the initial values ​​obtained from GA optimization, perform BP retraining. Update the BP parameters for the corresponding samples, and use Verification and monitoring were performed on the corresponding samples. The corresponding samples are not involved in loss calculation and backpropagation, but are only used for model evaluation to further optimize the weights and biases of the GA-BP network model.

[0041] Further, step S9 includes: S91. Using the finally trained GA-BP network model, perform forward prediction on the training set, validation set, and test set respectively to obtain the normalized prediction output: ,

[0042] S92. Based on the normalization relationship in step S26, perform inverse normalization to obtain the predicted value of the gas bearing capacity in terms of physical quantity:

[0043] The corresponding result is:

[0044]

[0045]

[0046] Composition of the predicted value set: ,

[0047] in, The training set is the set of predicted carrying capacity values. To validate the set of predicted bearing capacity values, This is the set of predicted bearing capacity values ​​for the test set.

[0048] Compared with the prior art, the present invention has the following advantages: 1. Addressing the strongly nonlinear mapping relationship between the bearing capacity of a hydrostatic toroidal throttling gas bearing and its structural and operating parameters, this invention optimizes the initial weights and biases of a backpropagation (BP) neural network using the Genetic Algorithm (GA) algorithm, introducing Xavier / Glorot weight boundary settings. Compared to fixed intervals, Xavier / Glorot boundaries are better suited to the network structure, reducing the risk of gradient vanishing and exploding. This allows the genetic algorithm to search for optimal initial weights and biases within a more reasonable parameter range, avoiding the uneven distribution of network activation values ​​that may occur with fixed interval initialization. This provides a good initial parameter distribution foundation for stable training of the neural network, thereby ensuring the stability and accuracy of gas bearing bearing capacity prediction. The model's prediction accuracy is significantly better than that of traditional BP neural networks, enabling more accurate capture of the changing patterns of gas bearing bearing capacity and achieving high-precision prediction.

[0049] 2. This invention introduces a hybrid fitness function, which simultaneously considers the training and validation set errors of gas bearing samples. During the genetic evolution process, it balances the model's fitting accuracy and generalization ability, effectively suppressing the overfitting tendency caused by traditional GA-BP focusing only on training errors and the problem that traditional genetic algorithms optimize neural networks by only pursuing the optimal solution of the training set and ignoring generalization performance. This makes the model more adaptable to unseen test data with different combinations of parameters such as design membrane thickness and gas supply pressure, and can maintain stable prediction results under complex working conditions. It significantly improves the applicability and reliability of the model under multiple working conditions and reduces the design iteration risk in gas bearing structure design.

[0050] 3. This invention significantly reduces the computational cost of fitness evaluation and accelerates the overall convergence speed of the genetic algorithm by introducing short-round backpropagation (BP) training in the genetic search phase to replace traditional long-round training. It effectively suppresses the technical problem of excessive computational overhead caused by long training rounds for each individual in traditional genetic algorithms for optimizing neural networks. This allows the method to maintain high computational efficiency when facing large-scale sample data, providing a feasible technical approach for the rapid optimization design of gas bearing structural parameters.

[0051] 4. This invention, based on the optimal initial weights and biases obtained after optimization by the genetic algorithm, performs joint constraint retraining on the training and validation sets to further optimize the neural network parameters and significantly improve prediction stability and accuracy. While maintaining the excellent initial characteristics obtained in the genetic stage, the model finely adjusts the network weights through sufficient backpropagation training, ultimately obtaining a prediction model with both high fitting accuracy and strong generalization ability. This provides reliable technical support for the accurate performance evaluation and structural optimization of hydrostatic gas bearings.

[0052] 5. This invention combines design sampling, network parameter configuration optimization, global search using a genetic algorithm, and backpropagation training of a neural network. Based on the design sampling method, representative samples covering the range of key parameters are efficiently obtained, rapidly establishing a high-precision nonlinear mapping model between the structural parameters, operating parameters, and load-bearing capacity of gas bearings. Compared with traditional methods that rely on complex test bench construction and large-scale numerical iterative calculations, this invention significantly shortens the performance evaluation cycle of gas bearings, significantly improves the efficiency of gas bearing structural optimization design, and simultaneously ensures high accuracy and strong generalization ability of the prediction results. It provides an efficient and reliable prediction method for the engineering application and optimization design of hydrostatic toroidal throttling gas bearings. Attached Figure Description

[0053] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0054] Figure 1 This is a flowchart of the method of the present invention.

[0055] Figure 2 This is a schematic diagram of a radial bearing structure provided in an embodiment of the present invention.

[0056] Figure 3 This is a diagram of a neural network training window provided in an embodiment of the present invention. Detailed Implementation

[0057] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. 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 should fall within the scope of protection of the present invention.

[0058] It should be noted that the terms "comprising" and "having" and any variations thereof in the specification, claims and accompanying drawings of this invention are intended to cover non-exclusive inclusion. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are explicitly listed, but may include other steps or units that are not explicitly listed or that are inherent to such processes, methods, products or devices.

[0059] like Figure 1 As shown, this invention provides a method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network, comprising: S1. Using a design sampling method, gas bearing characteristic data is generated within the preset range of structural and operating parameters. For each set of parameter combinations in the gas bearing characteristic data, the corresponding load-bearing capacity data is obtained through CFD numerical simulation or experimental testing. Multiple sets of sample datasets are constructed, and the sample datasets are divided into training set, validation set and test set according to a preset ratio. S2. Determine the network structure parameters, and use the gas bearing characteristic data as the network input and the bearing capacity data as the output to normalize the sample dataset. S3. Construct a BP neural network model, using normalized gas bearing feature data as input and normalized bearing capacity data as output. Train the network with different numbers of hidden layer nodes on the training set, and use the minimum mean square error of the validation set as the selection criterion for the number of hidden layer nodes. Train the constructed BP neural network model to determine the optimal number of hidden layer nodes, so as to ensure that the BP neural network model has the best fitting ability to the gas bearing bearing capacity data. S4. The weights and bias parameters of the BP neural network are used to form a set of parameters to be optimized, and the set of parameters to be optimized is encoded with real numbers to generate chromosome individuals for the genetic algorithm. S5. After determining the BP neural network structure, construct a GA-BP network model by combining the genetic algorithm; concatenate and encode the weight parameter segment and bias parameter segment of the network model into a chromosome individual in a preset order, and determine the value range of each parameter segment according to the Xavier / Glorot initialization rule, and set symmetrical upper and lower boundaries for each gene in the chromosome to constrain the chromosome search space. S6. Based on the generated chromosomes, construct a hybrid fitness function for the GA-BP network model to balance fitting accuracy and generalization ability. S7. Genetic operations are performed based on the hybrid fitness function. The elite retention strategy is used to retain the current best individual. The population is updated iteratively to obtain the chromosome with the best fitness. The optimal weight matrix and optimal bias vector of the corresponding GA-BP network model are then decoded. S8. Based on the weights and biases obtained from decoding, perform joint constraint retraining of the GA-BP network model using the training and validation sets to further optimize the weights and biases of the GA-BP network model. S9. Based on the further optimized weights and biases, forward prediction and inverse normalization are performed to finally obtain the predicted value of the hydrostatic gas bearing capacity.

[0060] In a specific implementation, as a preferred embodiment of the present invention, in step S1, the constructed sample dataset includes gas bearing feature data and load-bearing capacity data, wherein: The gas bearing characteristic data includes structural parameters and operating parameters, wherein the structural parameters include the orifice diameter. Number of throttle orifice rows Spacing between multiple rows of holes Number of throttling orifices End distance ratio Aspect Ratio Orifice length Design air film thickness Bearing inner diameter Operating parameters include gas supply pressure. Rotation speed steady-state load The characteristic data of the gas bearing are represented as follows:

[0061] in, Represents the characteristic data matrix of gas bearings. This indicates the total number of gas bearing samples. Representing feature dimension, Indicates the first Group gas bearing characteristic data; The bearing capacity data is represented as follows:

[0062] in, Represents a bearing capacity data vector. Indicates and The corresponding load-bearing capacity data.

[0063] like Figure 2As shown, the radial bearing structure used in this embodiment features two rows of throttling orifices, uniformly and symmetrically distributed along the bearing circumference. Latin hypercube sampling (LHS) is preferably used to determine the combination of operating and structural parameters of the gas bearing. LHS has the advantages of being suitable for multi-parameter problems, providing layered coverage, and exhibiting good space-filling properties, which can improve the uniform coverage of the parameter space with a given sample size. In this example, the model input only selects four dimensions: gas supply pressure, throttling orifice diameter, and design gas film thickness, with eccentricity representing the steady-state load. Based on the sampled parameter combinations, the corresponding load-bearing capacity data is obtained through CFD numerical simulation, constructing a dataset of 270 gas bearing load-bearing capacities.

[0064] In a specific implementation, as a preferred embodiment of the present invention, step S2 includes: S21. Let the number of nodes in the input layer be... , Consistent with the characteristic dimensions of gas bearings, let the number of output layer nodes be... The corresponding bearing capacity value; S22, Gas bearing feature data matrix Defined as input matrix , load-bearing capacity data vector Defined as output vector ; S23, Definition of the 3D features, as follows:

[0065] in, Indicates the first The sample at the th Values ​​on the dimensional features; Indicates the first in the training set The minimum value of a feature; Indicates the first in the training set The maximum value of a feature; S24, Based on the definition of the first The gas bearing feature data, including structural parameters and operating parameters, is used as training input features and linearly normalized. Each feature dimension is mapped to the [0,1] interval, and the normalization transformation yields:

[0066] in, express The normalized value; S25, Using the training set The minimum and maximum values ​​of the dimensional features are used to perform the same transformation on the gas bearing feature data of the validation and test sets to ensure consistent data scale. S26. Using the bearing capacity data as the training output, perform linear normalization, mapping each feature dimension to the [0,1] interval, and then normalizing to obtain:

[0067] in, Indicates the first The true label value of the nth sample, i.e., the nth Load-bearing capacity data corresponding to the group of gas bearing samples; This represents the minimum value of the training set labels; This represents the maximum value of the training set labels.

[0068] In a specific implementation, as a preferred embodiment of the present invention, step S3 includes: S31. Let the search range for the number of hidden layer nodes be:

[0069] in, Indicates the number of hidden layer nodes; S32, For each hidden layer node number Construct a single-hidden-layer feedforward backpropagation neural network model, including an input layer, a hidden layer, and an output layer, where the input layer contains... The hidden layer contains 10 neurons for receiving input features; The number of neurons (i.e., the number of hidden layer nodes obtained through the search); the output layer contains Each neuron outputs a prediction result; S33. Calculate the mean square error of the training set, using the following formula:

[0070] in, Indicates in The mean squared error of the training set with a number of hidden layer nodes; Indicates the number of samples in the training set; A set of indices representing the samples in the training set; Indicates the normalized i-th The true values ​​of the training samples in the training set; Represents the normalized first i Predicted values ​​for each training sample in the training set; S34. Use a BP neural network model to test the normalized validation set. Make predictions and obtain the results. Validation set prediction output with a number of hidden layer nodes ; S35. Calculate the mean square error of the verification set, using the following formula:

[0071] in, Indicates in The mean squared error of the validation set with a number of hidden layer nodes; Indicates the number of samples in the validation set; A set of indices representing the validation set samples; Indicates the normalized i-th The true values ​​of the training samples in the validation set; Indicates the normalized i-th Predicted values ​​of training samples in the validation set; S36. Select a value that makes the mean square error of the validation set... The minimum number of hidden layer nodes is found, and the optimal number of hidden layer nodes is as follows:

[0072] in, This represents the optimal number of nodes in the hidden layer.

[0073] In a specific implementation, as a preferred embodiment of the present invention, step S4 includes: S41. Suppose that the trained BP neural network model contains a weight matrix from the input layer to the hidden layer. ∈ (Association of gas bearing input parameters and hidden layer features), bias vector of the hidden layer ∈ Weight matrix from hidden layer to output layer ∈ (Associating hidden layer features with gas bearing load capacity data) and the bias vector of the output layer ∈ ; S42. Expand the weights and biases from step S41 into a real chromosome vector in a fixed order, as follows:

[0074] in, Indicates chromosome length, .

[0075] In a specific implementation, as a preferred embodiment of the present invention, step S5 includes: S51. Set the weight boundaries from the input layer to the hidden layer as follows: , ∈[- , ] S52. Set the offset boundary of the hidden layer as follows: , ∈[- , ] S53. Set the weight boundaries from the hidden layer to the output layer as follows: , ∈[- , ] S54. Set the offset boundary of the output layer as follows: , ∈[- , ] S55. The upper and lower boundaries of each gene are determined by the parameter boundary set. , , It is assembled by segmenting according to the sequence of chromosome gene combinations, and each gene corresponds to , , , One of the scalar parameter elements; S56. Set the genetic algorithm parameters, including the maximum number of generations. Population size (sizepop) and crossover probability and mutation probability In this implementation, the preferred option is... Take 100, sizepop takes 50. Take 0.7, The value is set to 0.04. Based on the defined boundary values ​​for each gene, an initial GA population is generated by randomly sampling each gene on the chromosome within the corresponding boundary interval. This ensures that the initial individuals meet the parameter scale constraints and provide a stable starting point for subsequent genetic searches.

[0076] S57, For each individual k =1, ..., sizepop, generate chromosomes by randomly sampling within the corresponding boundaries: ,

[0077] in, Indicates the first Individual chromosome vector The Middle The values ​​of each gene; Indicates the first The lower boundary of a gene; Indicates the first The upper boundary of a gene; This represents the random number used for random sampling. In this embodiment, each chromosome corresponds to a set of initial weights and biases for a GA-BP neural network model.

[0078] In a specific implementation, as a preferred embodiment of the present invention, step S6 includes: S61, regarding the first Chromosomes Decode to obtain , , and And assign these values ​​to the GA-BP network model as initial parameters of the GA-BP network model; S62, Training Set Perform short-round backpropagation training to obtain the predicted output set. In this embodiment, short-round BP training refers to performing a preset number of BP updates for each individual in the hybrid optimization of the GA-BP network model, so that the network parameters can be quickly adjusted locally while keeping the computational cost under control. Then, the performance of the network after short training is used as the basis for evaluating the individual fitness, so as to improve the discriminativeness of fitness evaluation and accelerate the overall convergence.

[0079] S63. Perform forward propagation on the validation set to obtain the validation set prediction output set. ; S64, Calculate the first The mean absolute error of the training set during BP training with a short number of chromosomes reflects the fitting accuracy of the GA-BP network model to the known gas bearing load capacity data. The formula is as follows: ,

[0080] in, Indicates the first The mean absolute error of the training set during short-round backpropagation training of chromosomes; This indicates the number of short-round backpropagation (BP) training cycles. Normalized predicted values ​​of training samples in the individual training set; S65, Calculate the first The mean absolute error of the validation set during training with short chromosome rounds of backpropagation (BP) reflects the model's ability to generalize to unseen gas bearing capacity data. The formula is as follows: ,

[0081] in, Indicates the first The mean absolute error on the validation set during BP training with a short number of chromosome rounds. Indicates the first round after short-round training Normalized predicted values ​​of training samples in the individual validation set; S66. Calculate the error of each individual in the population after short-round backpropagation training independently as the mixed fitness function:

[0082] in, Represents the mixed fitness function. The weight parameters for the validation set range from 0.1 to 0.3. In this embodiment... 0.3 is preferred.

[0083] In a specific implementation, as a preferred embodiment of the present invention, step S7 includes: S71. Select high-quality individuals based on the hybrid fitness function, and calculate the... The selection weights for each individual are calculated using the following formula:

[0084] in, Indicates the first The selection weights corresponding to each individual; This represents a small constant to prevent the denominator from being zero; S72, the first calculation-based The selection weight corresponding to each individual The probability of an individual's choice is calculated using the following formula:

[0085] S73, Using roulette wheel betting method Sizepop individuals are extracted from the original population to form a new generation population, retaining high-quality gas bearing parameters, namely the network weights and biases corresponding to the gas bearing characteristic data.

[0086] S74. Crossover of the parent individuals obtained through selection operations to generate offspring, achieving chromosomal gene recombination, specifically including: For the two parent individuals to be crossed and Randomly select a gene location ,make:

[0087]

[0088]

[0089] And Within the corresponding boundary range, high-quality genes are integrated; among them, Indicates the parent generation In the The values ​​of each gene locus; Indicates the parent generation In the The values ​​of each gene locus; The crossover coefficient is represented and follows a uniform distribution. These represent the two offspring in the [number]th generation. The values ​​of each gene locus; S75. Perform mutation operations on offspring individuals according to the mutation probability to increase population diversity, specifically including: For the probability of mutation Selected individuals and gene locations Define the step size factor that varies with the number of generations:

[0090] in, Indicates the variable-asynchronous length coefficient; Represents a random number that follows a uniform distribution; Indicates the current generation number; Indicates the maximum number of generations; If random number Then, it mutates upwards, resulting in... ( ) Otherwise, it mutates downwards, resulting in... ( ) and will Crop to boundary [ , Within this range, increasing population diversity helps avoid getting trapped in local optima; in the above formula, Indicates the chromosome before the mutation The One gene; This represents the new gene value obtained after the mutation; Indicates the first The lower boundary of a gene; Indicates the first The upper boundary of a gene; S76. Design elite retention strategy, as follows: At the end of each generation, compare the error of the best individual in the current generation. Error compared to historical best ,like Then update the best individual (the best individual in the iteration process, representing the best in the current generation; if iterates to the last generation, it is regarded as the globally best individual), and replace the individual with the largest error in the current population with the best individual to ensure that the network weights and biases with the smallest prediction error of the gas bearing capacity of the GA-BP network model are not lost after the genetic algorithm is optimized. S77, Based on maximum number of evolutions Repeat steps S71-S76 to obtain the chromosome with optimal fitness. ; S78, after completion After generations of evolution, based on the chromosome with the best fitness Decode the values ​​to obtain the optimal initial weights and biases for the corresponding BP network: , , ,

[0091] in, This represents the optimal weight matrix from the input layer to the hidden layer; This represents the optimal bias vector of the hidden layer; This represents the optimal weight matrix from the hidden layer to the output layer; This represents the optimal bias vector of the output layer.

[0092] In a specific implementation, as a preferred embodiment of the present invention, step S8 includes: S81. Concatenate the normalized training set, validation set, and test set: = , =

[0093] in, This represents the fully normalized gas bearing characteristic data. The test set is used only for the final forward prediction and performance evaluation, and does not participate in error backpropagation and parameter updates during the training process, as it contains fully normalized bearing capacity data. S82, Set the training set, validation set, and test set respectively. Index range in:

[0094]

[0095]

[0096] S83. Based on the initial values ​​obtained from GA optimization, perform BP retraining. Update the BP parameters for the corresponding samples, and use Verification and monitoring were performed on the corresponding samples. The corresponding samples are not involved in loss calculation and backpropagation, but are only used for model evaluation to further optimize the weights and biases of the GA-BP network model.

[0097] In a specific implementation, as a preferred embodiment of the present invention, step S9 includes: S91. Using the finally trained GA-BP network model, perform forward prediction on the training set, validation set, and test set respectively to obtain the normalized prediction output: ,

[0098] S92. Based on the normalization relationship in step S26, perform inverse normalization to obtain the predicted value of the gas bearing capacity in terms of physical quantity:

[0099] The corresponding result is:

[0100]

[0101]

[0102] Composition of the predicted value set: ,

[0103] in, The training set is the set of predicted carrying capacity values. To validate the set of predicted bearing capacity values, This is the set of predicted bearing capacity values ​​for the test set.

[0104] Example To verify the authenticity and effectiveness of the method of the present invention, the error and accuracy indicators were calculated, and the process is as follows: The training set error vector is calculated using the following formula:

[0105] in, This represents the normalized vector of true values ​​from the training set label data. This represents the vector of predicted values ​​for the label data in the training set after normalization.

[0106] The average error of the training set is calculated using the following formula:

[0107] in, Indicates the training set number The error of each sample.

[0108] The mean absolute error of the training set is calculated using the following formula:

[0109] The validation set error vector is calculated using the following formula:

[0110] in, This represents the normalized vector of true values ​​for the validation set label data. This represents the vector of predicted values ​​for the validation set label data after normalization.

[0111] The average error of the validation set is calculated using the following formula:

[0112] in, Indicates the verification set number The error of each sample.

[0113] The mean absolute error of the validation set is calculated using the following formula:

[0114] The test set error vector is calculated using the following formula:

[0115] in, This represents the normalized vector of true values ​​for the test set label data. This represents the vector of predicted values ​​for the test set label data after normalization.

[0116] The average error of the test set is calculated using the following formula:

[0117] in, Representative test set number The error of each sample.

[0118] The mean absolute error of the test set is calculated using the following formula:

[0119] The relative error of the training set is calculated using the following formula:

[0120] in, Indicates the training set number The prediction error for each sample; Indicates the training set number The true carrying capacity of each sample.

[0121] The mean absolute percentage error of the training set is calculated using the following formula:

[0122] The relative error of the validation set is calculated using the following formula:

[0123] in, Indicates the verification set number The prediction error for each sample; Indicates the verification set number The true carrying capacity of each sample.

[0124] The mean absolute percentage error of the validation set is calculated using the following formula:

[0125] The relative error of the test set is calculated using the following formula:

[0126] in, Indicates the test set number The prediction error for each sample; Indicates the test set number The true carrying capacity of each sample.

[0127] The mean absolute percentage error of the test set is calculated using the following formula:

[0128] It reflects the proportion of the deviation between the predicted value and the actual value of the gas bearing.

[0129] The root mean square error of the training set is calculated using the following formula:

[0130] The root mean square error of the validation set is calculated using the following formula:

[0131] The root mean square error of the test set is calculated using the following formula:

[0132] The coefficient of determination for the training set is calculated using the following formula:

[0133] in, This represents the mean of the true values ​​in the training set. ; The formula for calculating the coefficient of determination on the validation set is as follows:

[0134] in, This represents the mean of the true values ​​in the validation set. ; The coefficient of determination for the test set is calculated using the following formula:

[0135] in, This represents the mean of the true values ​​in the test set. .

[0136] The closer the value is to 1, the stronger the model's ability to explain the load-bearing capacity of gas bearings.

[0137] Pearson correlation coefficient between predicted and true values The standard covariance formula is used to calculate the linear correlation between the predicted and actual values ​​of the gas bearing, where: The formula for calculating the sample mean of the true values ​​in the training set is as follows:

[0138] The sample mean of the predicted values ​​in the training set is calculated using the following formula:

[0139] The sample covariance between the actual values ​​and the predicted values ​​in the training set is calculated using the following formula:

[0140] The sample standard deviation of the true values ​​in the training set is calculated using the following formula:

[0141] The sample standard deviation of the predicted values ​​in the training set is calculated using the following formula:

[0142] The Pearson correlation coefficient between the training values ​​and the true values ​​in the training set is calculated using the following formula:

[0143] The formula for calculating the sample mean of the true values ​​on the validation set is as follows:

[0144] The sample mean of the predicted values ​​on the validation set is calculated using the following formula:

[0145] The sample covariance between the actual values ​​and the predicted values ​​on the validation set is calculated using the following formula:

[0146] The sample standard deviation of the true values ​​on the validation set is calculated using the following formula:

[0147] The sample standard deviation of the predicted values ​​on the validation set is calculated using the following formula:

[0148] The Pearson correlation coefficient between the training values ​​and the true values ​​on the validation set is calculated using the following formula:

[0149] The formula for calculating the sample mean of the true values ​​in the test set is as follows:

[0150] The sample mean of the predicted values ​​for the test set is calculated using the following formula:

[0151] The sample covariance between the actual values ​​and predicted values ​​in the test set is calculated using the following formula:

[0152] The sample standard deviation of the true values ​​in the test set is calculated using the following formula:

[0153] The sample standard deviation of the predicted values ​​for the test set is calculated using the following formula:

[0154] The Pearson correlation coefficient between the training values ​​and the true values ​​in the test set is calculated using the following formula:

[0155] like Figure 3 As shown, open the training window and view the linear correlation curves between the actual and predicted values ​​of the gas bearing capacity training set, validation set, and test set. This is used to intuitively evaluate the fitting and generalization performance of the GA-BP model.

[0156] In this embodiment, the labeled dataset is divided into a training set, a validation set, and a test set in a 7:1.5:1.5 ratio. The training and validation sets are input into the improved network model for iterative training to obtain the optimal model weights. This invention successfully obtained the optimal model weights by iteratively training the original BP model. Through model regression analysis, regression fitting was successfully achieved, and the fitting results are as follows. Figure 3As shown, this invention uses a GA-BP neural network, employing operations such as crossover and mutation to find the optimal weight bias, thereby reducing the prediction error of the BP neural network and improving the generalization ability of the prediction model. The GA-BP neural network exhibits stronger generalization ability and significantly improved fitting performance. The comparison results are shown in the table below:

[0157] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network, characterized in that, include: S1. Using a design sampling method, gas bearing characteristic data is generated within the preset range of structural and operating parameters. For each set of parameter combinations in the gas bearing characteristic data, the corresponding load-bearing capacity data is obtained through CFD numerical simulation or experimental testing. Multiple sets of sample datasets are constructed, and the sample datasets are divided into training set, validation set and test set according to a preset ratio. S2. Determine the network structure parameters, and use the gas bearing characteristic data as the network input and the bearing capacity data as the output to normalize the sample dataset. S3. Construct a BP neural network model, using normalized gas bearing feature data as input and normalized bearing capacity data as output. Train the network with different numbers of hidden layer nodes on the training set, and use the minimum mean square error of the validation set as the selection criterion for the number of hidden layer nodes. Train the constructed BP neural network model to determine the optimal number of hidden layer nodes, so as to ensure that the BP neural network model has the best fitting ability to the gas bearing bearing capacity data. S4. The weights and bias parameters of the BP neural network are used to form a set of parameters to be optimized, and the set of parameters to be optimized is encoded with real numbers to generate chromosome individuals for the genetic algorithm. S5. After determining the BP neural network structure, construct a GA-BP network model by combining the genetic algorithm; concatenate and encode the weight parameter segment and bias parameter segment of the network model into a chromosome individual in a preset order, and determine the value range of each parameter segment according to the Xavier / Glorot initialization rule, and set symmetrical upper and lower boundaries for each gene in the chromosome to constrain the chromosome search space. S6. Based on the generated chromosomes, construct a hybrid fitness function for the GA-BP network model to balance fitting accuracy and generalization ability. S7. Genetic operations are performed based on the hybrid fitness function. The elite retention strategy is used to retain the current best individual. The population is updated iteratively to obtain the chromosome with the best fitness. The optimal weight matrix and optimal bias vector of the corresponding GA-BP network model are then decoded. S8. Based on the weights and biases obtained from decoding, perform joint constraint retraining of the GA-BP network model using the training and validation sets to further optimize the weights and biases of the GA-BP network model. S9. Based on the further optimized weights and biases, forward prediction and inverse normalization are performed to finally obtain the predicted value of the hydrostatic gas bearing capacity.

2. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, In step S1, the constructed sample dataset includes gas bearing feature data and load-bearing capacity data, wherein: The gas bearing characteristic data includes structural parameters and operating parameters. The structural parameters include orifice diameter, number of orifice rows, spacing between rows, number of orifices, end-to-end distance ratio, length-to-diameter ratio, orifice length, design gas film thickness, and bearing inner diameter. The operating parameters include supply pressure, rotational speed, and steady-state load. The gas bearing characteristic data can be expressed as follows: in, Represents the characteristic data matrix of gas bearings. This indicates the total number of gas bearing samples. Representing feature dimension, Indicates the first Group gas bearing characteristic data; The bearing capacity data is represented as follows: in, Represents a bearing capacity data vector. Indicates and The corresponding load-bearing capacity data.

3. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S2 includes: S21. Let the number of nodes in the input layer be... , Consistent with the characteristic dimensions of gas bearings, let the number of output layer nodes be... The corresponding bearing capacity value; S22, Gas bearing feature data matrix Defined as input matrix , load-bearing capacity data vector Defined as output vector ; S23, Definition of the 3D features, as follows: in, Indicates the first The sample at the th Values ​​on the dimensional features; Indicates the first in the training set The minimum value of a feature; Indicates the first in the training set The maximum value of a feature; S24, Based on the definition of the first The gas bearing feature data, including structural parameters and operating parameters, is used as training input features and linearly normalized. Each feature dimension is mapped to the [0,1] interval, and the normalization transformation yields: in, express The normalized value; S25, Using the training set The minimum and maximum values ​​of the dimensional features are used to perform the same transformation on the gas bearing feature data of the validation and test sets to ensure consistent data scale. S26. Using the bearing capacity data as the training output, perform linear normalization, mapping each feature dimension to the [0,1] interval, and then normalizing to obtain: in, Indicates the first The true label value of the nth sample, i.e., the nth Load-bearing capacity data corresponding to the group of gas bearing samples; This represents the minimum value of the training set labels; This represents the maximum value of the training set labels.

4. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S3 includes: S31. Let the search range for the number of hidden layer nodes be: in, Indicates the number of hidden layer nodes; S32, For each hidden layer node number Construct a single-hidden-layer feedforward backpropagation neural network model, including an input layer, a hidden layer, and an output layer, where the input layer contains... The hidden layer contains 10 neurons for receiving input features; 10 neurons; the output layer contains 100 neurons; Each neuron outputs a prediction result; S33. Calculate the mean squared error of the training set, using the following formula: in, Indicates in The mean squared error of the training set with a number of hidden layer nodes; Indicates the number of samples in the training set; A set of indices representing the samples in the training set; Indicates the normalized i-th The true values ​​of the training samples in the training set; Represents the normalized first i Predicted values ​​for each training sample in the training set; S34. Use a BP neural network model to test the normalized validation set. Make predictions and obtain the results. Validation set prediction output with a number of hidden layer nodes ; S35. Calculate the mean square error of the verification set, using the following formula: in, Indicates in The mean squared error of the validation set with a number of hidden layer nodes; Indicates the number of samples in the validation set; A set of indices representing the validation set samples; Indicates the normalized i-th The true values ​​of the training samples in the validation set; Indicates the normalized i-th Predicted values ​​of training samples in the validation set; S36. Select a value that makes the mean square error of the validation set... The minimum number of hidden layer nodes is found, and the optimal number of hidden layer nodes is as follows: in, This represents the optimal number of nodes in the hidden layer.

5. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S4 includes: S41. Suppose that the trained BP neural network model contains a weight matrix from the input layer to the hidden layer. ∈ Bias vector of hidden layer ∈ Weight matrix from hidden layer to output layer ∈ and the bias vector of the output layer ∈ ; S42. Expand the weights and biases from step S41 into a real chromosome vector in a fixed order, as follows: in, Indicates chromosome length, .

6. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S5 includes: S51. Set the weight boundaries from the input layer to the hidden layer as follows: ， ∈[- ， ] S52. Set the offset boundary of the hidden layer as follows: ， ∈[- ， ] S53. Set the weight boundaries from the hidden layer to the output layer as follows: ， ∈[- ， ] S54. Set the offset boundary of the output layer as follows: ， ∈[- ， ] S55. The upper and lower boundaries of each gene are determined by the parameter boundary set. , , It is assembled by segmenting according to the sequence of chromosome gene combinations, and each gene corresponds to , , , One of the scalar parameter elements; S56. Set the genetic algorithm parameters, including the maximum number of generations, population size, crossover probability, and mutation probability; S57, For each individual k =1, ..., sizepop, generate chromosomes by randomly sampling within the corresponding boundaries: ， in, Indicates the first Individual chromosome vector The Middle The values ​​of each gene; Indicates the first The lower boundary of a gene; Indicates the first The upper boundary of a gene; This represents the random number used for random sampling.

7. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S6 includes: S61, regarding the first Decoding each chromosome yields... , , and And assign these values ​​to the GA-BP network model as initial parameters of the GA-BP network model; S62, Training Set Perform short-round backpropagation training to obtain the predicted output set. ; S63. Perform forward propagation on the validation set to obtain the validation set prediction output set. ; S64, Calculate the first The mean absolute error of the training set during BP training with a short number of chromosomes reflects the fitting accuracy of the GA-BP network model to the known gas bearing load capacity data. The formula is as follows: ， in, Indicates the first The mean absolute error of the training set during short-round backpropagation training of individual units; This indicates the number of short-round backpropagation (BP) training cycles. Normalized predicted values ​​of training samples in the individual training set; S65, Calculate the first The mean absolute error of the validation set during training of individual short-wheel-count backpropagation (BP) models reflects the model's ability to generalize to unseen gas bearing capacity data. The formula is as follows: ， in, Indicates the first The mean absolute error of the validation set during short-round backpropagation training of each individual Indicates the first round after short-round training Normalized predicted values ​​of training samples in the individual validation set; S66. Calculate the error of each individual in the population after short-round backpropagation training independently as the mixed fitness function: in, Represents the mixed fitness function. The weight parameters for the validation set.

8. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S7 includes: S71. Select high-quality individuals based on the hybrid fitness function, and calculate the... The selection weights for each individual are calculated using the following formula: in, Indicates the first The selection weights corresponding to each individual; This represents a constant to prevent the denominator from being zero. S72, the calculation-based first The selection weight corresponding to each individual The probability of an individual's choice is calculated using the following formula: S73, Using roulette wheel betting method Sizepop individuals are extracted from the original population to form a new generation population, while retaining high-quality gas bearing parameters, namely the network weights and biases corresponding to the gas bearing feature data. S74. Crossover of the parent individuals obtained through selection operations to generate offspring, achieving chromosomal gene recombination, specifically including: For the two parent individuals to be crossed and Randomly select a gene location ,make: And Within the corresponding boundary range, high-quality genes are integrated; among them, Indicates the parent generation In the The values ​​of each gene locus; Indicates the parent generation In the The values ​​of each gene locus; The crossover coefficient is represented and follows a uniform distribution. These represent the two offspring in the [number]th generation. The values ​​of each gene locus; S75. Perform mutation operations on offspring individuals according to the mutation probability, specifically including: For the probability of mutation Selected individuals and gene locations Define the step size factor that varies with the number of generations: in, Indicates the variable-asynchronous length coefficient; Represents a random number that follows a uniform distribution; Indicates the current generation number; Indicates the maximum number of generations; If random number Then, it mutates upwards, resulting in... ( ) Otherwise, it will mutate downwards to obtain... ( ) and will Crop to boundary [ , Within this range, increasing population diversity helps avoid getting trapped in local optima; in the above formula, Indicates the chromosome before the mutation The One gene; This represents the new gene value obtained after the mutation; Indicates the first The lower boundary of a gene; Indicates the first The upper boundary of a gene; S76. Design elite retention strategy, as follows: At the end of each generation, compare the error of the best individual in the current generation. Error compared to historical best ,like If the optimal individual is updated, the individual with the largest error in the current population is replaced with the optimal individual to ensure that the network weights and biases with the smallest prediction error of the gas bearing capacity of the GA-BP network model are not lost after the genetic algorithm is optimized. S77, Based on maximum number of evolutions Repeat steps S71-S76 to obtain the chromosome with the best fitness. ; S78, after completion After generations of evolution, based on the chromosome with the best fitness Decoding yields the optimal initial weight matrix and bias vector for the corresponding BP network: ， ， ， in, This represents the optimal weight matrix from the input layer to the hidden layer; This represents the optimal bias vector of the hidden layer; This represents the optimal weight matrix from the hidden layer to the output layer; This represents the optimal bias vector of the output layer.

9. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S8 includes: S81. Concatenate the normalized training set, validation set, and test set: = ， = in, This represents the fully normalized gas bearing characteristic data. The test set is used only for the final forward prediction and performance evaluation, and does not participate in error backpropagation and parameter updates during the training process, as it contains fully normalized bearing capacity data. S82, Set the training set, validation set, and test set respectively. Index range in: S83. Based on the initial values ​​obtained from GA optimization, perform BP retraining. Update the BP parameters for the corresponding samples, and use Verification and monitoring were performed on the corresponding samples. The corresponding samples are not involved in loss calculation and backpropagation, but are only used for model evaluation to further optimize the weights and biases of the GA-BP network model.

10. The method for predicting the bearing capacity of a hydrostatic toroidal throttling gas bearing based on a GA-BP neural network according to claim 1, characterized in that, Step S9 includes: S91. Using the finally trained GA-BP network model, perform forward prediction on the training set, validation set, and test set respectively to obtain the normalized prediction output: ， S92. Based on the normalization relationship in step S26, perform inverse normalization to obtain the predicted value of the gas bearing capacity in terms of physical quantity: The corresponding result is: Composition of the predicted value set: ， in, The training set is the set of predicted carrying capacity values. To validate the set of predicted bearing capacity values, This is the set of predicted bearing capacity values ​​for the test set.

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