A low-frequency EIS rapid prediction method for lithium-ion batteries

Abstract

The application relates to a lithium ion battery low-frequency EIS rapid prediction method, which comprises the following steps: acquiring a lithium ion battery EIS training sample; constructing a physical constraint condition variational autoencoder; constructing a total loss function composed of a reconstruction loss, a KL divergence loss and a physical constraint loss and training the physical constraint condition variational autoencoder; determining an optimal physical constraint condition variational autoencoder configuration by adopting Bayesian optimization; collecting a medium-high frequency EIS of a lithium ion battery to be measured, inputting the medium-high frequency EIS into the optimal configuration physical constraint condition variational autoencoder to obtain a corresponding low-frequency EIS prediction sequence; and splicing the medium-high frequency EIS measured sequence and the low-frequency EIS prediction sequence according to a frequency sequence to obtain a full-frequency EIS of the lithium ion battery to be measured. Compared with the prior art, the application has the advantages of short time consumption, effective inhibition of local fluctuation, high precision and stability of the model, high parameter configuration efficiency and the like.

Description

Technical Field

[0001] This invention relates to the field of battery energy storage state estimation, and in particular to a method for rapid prediction of low-frequency EIS of lithium-ion batteries. Background Technology

[0002] Electrochemical induction (EIS) is a non-destructive frequency domain characterization method. By applying a small AC perturbation to the battery and measuring the voltage and current responses at different frequencies, it can separate electrochemical processes with different time scales, such as ohmic conduction, interface film, charge transfer, concentration polarization, and solid-phase diffusion. Compared with DC testing, which only reflects the overall response, EIS can provide richer information on the battery's internal dynamics and aging.

[0003] Low-frequency EIS primarily characterizes slow processes such as lithium-ion diffusion and concentration polarization, and its characteristics are closely related to battery capacity decay, interface deposition, loss of active materials, and limited mass transfer. However, the lower the frequency, the longer it takes to complete one or more perturbation cycles, and the measurement time for full-frequency point-by-point sweep is usually mainly consumed at low-frequency points.

[0004] Dynamic EIS can directly measure impedance during battery charging or discharging, showing potential for online applications. However, during dynamic operation, the battery's state of charge, concentration gradient, and polarization voltage continuously change over time. The long period of low-frequency disturbances causes the low-frequency response to deviate from the quasi-steady-state assumption, easily leading to trajectory distortion, reverse bending, and characteristic drift. Directly using this type of low-frequency data for tasks such as SOH estimation or DRT inversion reduces the stability and reliability of the results.

[0005] Existing methods for shortening EIS measurement time mainly include reducing low-frequency sampling density, using multiple sinusoidal or pseudo-random excitations, directly estimating impedance from current and voltage transients, and reconstructing full-frequency EIS using ordinary neural networks. Reducing sampling density will result in the loss of diffusion process information; broadband excitation and transient identification still require dedicated excitation and signal processing; ordinary data-driven models usually only minimize numerical errors and lack constraints on the continuity of the EIS curve, the trend of low-frequency real part changes, and the morphology of the diffusion tail, which can easily lead to local oscillations or non-physical predictions under various temperature, cell, and dynamic operating conditions.

[0006] A search of Chinese Patent Publication No. CN121918015A reveals a rapid detection method and system for full-band electrochemical impedance spectroscopy of lithium batteries. The method first acquires the Bode plot of the lithium battery's electrochemical impedance spectroscopy, identifying and extracting characteristic frequency points representing the transitions of different electrochemical processes. Then, based on these characteristic frequency points, an EK-DIBS excitation sequence with concentrated spectral energy at the characteristic frequency points is constructed using a discrete-interval binary sequence method. After applying this excitation to the battery, voltage and current response signals are simultaneously acquired, and the impedance values ​​corresponding to the characteristic frequency points are extracted using a fast Fourier transform. Finally, these impedance values, along with battery operating parameters, are input into a pre-trained Sequence Attention Kernel (SeqKAN) network model to reconstruct the complete full-band electrochemical impedance spectroscopy. However, the existing patent requires a hardware excitation circuit, a synchronous acquisition module, and complex operating conditions, resulting in high hardware costs and difficulty in online implementation. In addition, the existing patent requires pre-obtaining complete EIS diagrams under different SOCs to draw Bode plots, calculating phase angle derivatives to find extreme points, then combining electrochemical mechanisms to screen candidate frequency points, and finally using SHAP interpretability analysis and MAE index to screen the optimal frequency point combination, which is too cumbersome. Furthermore, the existing patent does not introduce any physical prior constraints in the field of electrochemistry, and the prediction results are prone to non-physical oscillations.

[0007] Therefore, there is a need for a rapid full-frequency EIS acquisition method that can generate low-frequency EIS based on rapidly obtained mid-to-high frequency EIS, takes into account numerical accuracy and electrochemical-physical consistency, and is applicable to steady-state and dynamic operating conditions as well as different battery systems. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for rapid prediction of low-frequency EIS in lithium-ion batteries.

[0009] The objective of this invention can be achieved through the following technical solutions: A method for rapid prediction of low-frequency EIS in lithium-ion batteries includes: Step S1: Obtain the full-frequency EIS of the lithium-ion battery under different states of charge, temperatures and operating conditions as training samples, where EIS is electrochemical impedance spectroscopy; arrange the real and imaginary parts of the impedance at each frequency point of the training samples in order from high to low frequency, and divide the full-frequency EIS into a mid-to-high frequency input segment and a low-frequency target segment. Step S2: Construct a variational autoencoder with physical constraints; Step S3: Construct a total loss function consisting of reconstruction loss, KL divergence loss, and physical constraint loss, and train the physical constraint variational autoencoder using the training samples; Step S4: Use Bayesian optimization to determine the optimal physical constraint variational autoencoder configuration; Step S5: Acquire the mid-to-high frequency EIS of the lithium-ion battery under test, and input the mid-to-high frequency EIS of the lithium-ion battery under test into the optimally configured physical constraint variational autoencoder in step S4 to obtain the low frequency EIS prediction sequence of the lithium-ion battery under test. Step S6: The mid-to-high frequency EIS measured sequence of the lithium-ion battery under test and the low frequency EIS predicted sequence of the lithium-ion battery under test are spliced ​​together in frequency order to obtain the full frequency EIS of the lithium-ion battery under test.

[0010] As a preferred technical solution, the length of the mid-to-high frequency input segment is adjusted according to the requirements of measurement efficiency and prediction accuracy, and its proportion of the total number of full-frequency EIS frequency points is 40% to 80%, with the remaining frequency points constituting the low-frequency target segment.

[0011] As a preferred technical solution, the EIS includes steady-state EIS and dynamic EIS acquired under different charging states; wherein, the steady-state EIS is impedance data acquired after the battery has been left to stand and reached a relatively stable state, and the dynamic EIS is impedance data acquired directly during battery operation without being left to stand; the physical constraint variational autoencoder is suitable for low-frequency band prediction of steady-state EIS and dynamic EIS under different charging states.

[0012] As a preferred technical solution, the physical constraint variational autoencoder includes a recurrent neural network encoder and a condition mapping module; The recurrent neural network encoder is used to extract frequency sequence features from the mid-to-high frequency input segment; the conditional mapping module is used to generate conditional vectors.

[0013] As a preferred technical solution, the physical constraint variational autoencoder further includes a latent variable generation module and a recurrent neural network autoregressive decoder; the latent variable generation module is used to obtain latent variables; the recurrent neural network autoregressive decoder generates low-frequency EIS point by point according to the condition vector, latent variables and impedance information of the previous frequency point.

[0014] As a preferred technical solution, the recurrent neural network autoregressive decoder initializes the decoder hidden state by concatenating the latent variables and the condition vector, and when generating the current low-frequency point, it uses the predicted impedance of the previous frequency point and the condition vector as input together; during the training phase of the physical constraint variational autoencoder, the actual impedance of the previous frequency point is used to replace the predicted impedance with a preset probability, and the probability is gradually adjusted as the training process progresses.

[0015] As a preferred technical solution, the physical constraint loss includes second-order difference smoothing constraint, impedance real part monotonic constraint, and low-frequency diffusion tail stabilization constraint. The second-order difference smoothing constraint is used to limit the local curvature variation between adjacent low-frequency points; The monotonic constraint on the real part of the impedance is used to limit the non-physical decrease of the real part of the impedance as the frequency decreases.

[0016] As a preferred technical solution, the physical constraint loss also includes low-frequency diffusion tail stabilization constraint; The low-frequency diffusion tail stability constraint is used to limit the dispersion of the difference vector between adjacent frequency points at the low-frequency end.

[0017] As a preferred technical solution, the reconstruction loss includes impedance real part error, impedance imaginary part error and impedance magnitude error; the KL divergence loss is used to make the latent variable posterior distribution of the encoder output close to the preset Gaussian prior distribution; the weight of the KL divergence loss in the total loss function is gradually increased from a small value in the early stage of training.

[0018] As a preferred technical solution, step S4 specifically includes: configuring the recurrent neural network type, hidden dimension, number of layers, latent variable dimension, true input probability, learning rate, batch size, and number of training rounds of the encoder and decoder. , Using α, β, and γ as candidate variables, and the validation set RMSE, MAPE, and coefficient of determination as objective functions, the next set of candidate parameters is iteratively selected to obtain the optimal physical constraint variational autoencoder.

[0019] Compared with the prior art, the present invention has the following beneficial effects: 1. Significantly shortens the full-frequency EIS acquisition time, reducing the need for static operation under dynamic conditions. This invention only requires measuring the relatively short-duration mid-to-high frequency EIS and directly generates the low-frequency EIS prediction sequence using a trained variational autoencoder with physical constraints, avoiding the lengthy process of completing all low-frequency perturbation cycles point by point in traditional methods. Under dynamic conditions, this method can also reduce the need for long-term battery resting to eliminate polarization and meet the quasi-steady-state assumption, thereby significantly improving the acquisition efficiency of full-frequency EIS.

[0020] 2. Effectively suppresses local fluctuations and enhances spectral morphology modeling capabilities. This invention treats the EIS as a complex impedance sequence with continuously evolving characteristics along the frequency axis and constructs a model structure that includes a recurrent neural network encoder and an autoregressive decoder. The encoder processes the high-frequency input sequence point by point to extract the global spectral trend, while the decoder uses the predicted impedance of the previous frequency point and the conditional vector as input when generating low-frequency points. This sequence modeling method can simultaneously capture the global spectral shape and the continuous dependence between adjacent frequency points, and is more effective in suppressing local fluctuations in the prediction curve compared to ordinary fully connected networks.

[0021] 3. Ensure the physical consistency of the prediction results and reduce the impact of non-physical distortions. In the total loss function for model training, in addition to the usual reconstruction loss and KL divergence loss, three physical constraint losses are also jointly introduced: Second-order difference smoothing constraint: restricts local curvature changes between adjacent low-frequency points, suppressing spikes and oscillations inconsistent with the continuous electrochemical response; Monotonic constraint on the real part of impedance: restricts the non-physical decrease of the real part of impedance as the frequency decreases, which conforms to the physical law that the slow polarization process gradually manifests. Low-frequency diffusion tail stability constraint: Limit the dispersion of the differential vector between adjacent frequency points at the low-frequency end to maintain the stable extension trend of the diffusion tail.

[0022] These constraints not only make the prediction results numerically close to the measured values, but also satisfy the basic physical characteristics of low-frequency EIS, thereby reducing the negative impact of dynamic low-frequency distortion and non-physical oscillations of the pure data-driven model on subsequent battery state analysis.

[0023] 4. Improve model accuracy, stability, generalization ability, and parameter configuration efficiency. This invention utilizes the latent space of a conditional variational autoencoder to describe low-frequency response changes caused by factors such as different cycle stages, temperature, and individual differences. Furthermore, compared to existing patents, this invention can simultaneously handle steady-state EIS and dynamic EIS, enhancing the model's generalization ability. Simultaneously, it employs a Bayesian optimization method to jointly determine various hyperparameters and loss weights, including the recurrent neural network type, hidden dimension, number of layers, latent variable dimension, true input probability, learning rate, batch size, number of training epochs, and KL divergence loss weights and physical constraint loss weights. Using the prediction error on the validation set as the optimization objective, it avoids the tedious manual trial-and-error process, achieving synergistic optimization of the model structure and physical constraint weights, thereby improving the model's accuracy, stability, generalization ability, and parameter configuration efficiency. Attached Figure Description

[0024] Figure 1 This is the overall flowchart of the present invention; Figure 2 This is a schematic diagram of the physical constraint variational autoencoder of the present invention. Detailed Implementation

[0025] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0026] Example like Figure 1The diagram shown is the overall flowchart of the low-frequency EIS fast prediction and full-frequency reconstruction method of the present invention. Step S1: Obtain the full-frequency EIS of the lithium-ion battery under different cycle stages, state of charge, temperature, and steady-state or dynamic operating conditions. Each EIS contains multiple sampling points arranged from high to low frequency, and each sampling point consists of the real part and the imaginary part of the impedance.

[0027] The input ratio was set to multiple candidate values, including 40%, 50%, 60%, 70%, and 80%, and the low-frequency EIS prediction accuracy and corresponding measurement time were evaluated under different input ratios. After comprehensively comparing the prediction error and measurement efficiency, the first 60% of high-frequency to mid-frequency points were determined as the input segment, and the remaining 40% of mid-low-frequency to low-frequency points were determined as the prediction target segment.

[0028] Let the mid-to-high frequency input sequence be: The low-frequency target sequence is: in, and These represent the number of frequency points in the mid-to-high frequency input segment and the low-frequency target segment, respectively. and These represent the real and imaginary parts of the complex impedance, respectively.

[0029] Step S2: Construct a variational autoencoder with physical constraints. For example... Figure 2 The diagram shown is a schematic representation of the physical constraint variational autoencoder of the present invention.

[0030] The encoder includes a recurrent neural network encoder, a condition mapping module, a latent variable generation module, and a recurrent neural network autoregressive decoder.

[0031] Suppose that the medium-to-high frequency EIS input sequence contains The impedance points are arranged in descending order of frequency. The high-frequency impedance points are represented as follows: in, Represents the first in the mid-to-high frequency EIS input sequence One frequency point; Indicates the frequency point number within the mid-to-high frequency input range; This indicates the total number of frequency points contained in the mid-to-high frequency input segment; and They represent the first The real and imaginary parts of the complex impedance at each frequency point.

[0032] The encoder processes the input sequence point by point in frequency order, and its hidden state is updated as follows: in, For the first The high-frequency impedance point, This is the encoder hidden state corresponding to the previous frequency point. When the encoder completes all... After processing each frequency point, When the final hidden state is obtained ,Right now: It is a compressed representation of the entire mid-to-high frequency EIS input sequence, used to comprehensively characterize the overall impedance variation trend along the frequency direction and the local correlation information between adjacent frequency points. The conditional mapping module maps the final hidden state into a conditional vector. : in, This represents the weight matrix of the conditional mapping module; represents the bias vector of the conditional mapping module; tanh represents the hyperbolic tangent activation function, used to perform nonlinear transformations on the linear mapping results.

[0033] The encoder outputs the mean vector of the latent variable distribution through two independent linear mappings. Sum of logarithmic variance vector And obtain latent variables through reparameterization. : Here, ⊙ represents element-wise multiplication. A standard Gaussian random vector. Latent variables. Used to describe the varying factors in low-frequency responses that are not fully expressed by deterministic conditional vectors.

[0034] latent variables and condition vector By concatenating the features, the decoder initialization features are obtained. : Decoder initial hidden state for: Suppose the low-frequency EIS to be predicted includes The frequency point. In generating the first frequency point. When dealing with low-frequency impedance points, combine the impedance information from the previous frequency point with the condition vector. Together they serve as the decoder input: in, This indicates that the decoder generates the first... Input vector at a low-frequency impedance point; This indicates that the model generated in the previous step... A low-frequency impedance point.

[0035] Decoder hidden state and the predicted value at the current frequency point They are respectively: in, It includes the predicted real and imaginary parts of the impedance at the current low-frequency point. By using the previous frequency point as a recursive input, the model can generate complete low-frequency data using the continuous relationship between adjacent frequency points.

[0036] To reduce the accumulation of errors in autoregressive decoding, a real-value input adjustment strategy is adopted during the training phase: using probability... Input the actual impedance at the previous frequency point into the decoder, with probability The predicted impedance from the previous frequency point is used. A higher true input probability is used initially during training to provide stable guidance for the decoder; this is gradually reduced as training progresses. This allows the model to gradually acquire independent autoregressive generation capabilities.

[0037] Step S3: Construct a total loss function consisting of reconstruction loss, KL divergence loss, and physical constraint loss, and train the encoder.

[0038] For length of Predicting low-frequency sequences, second-order difference smoothing constraint for: This constraint limits abrupt changes in local curvature and suppresses spikes and oscillations in the predicted curve that are inconsistent with the continuous electrochemical response.

[0039] When frequencies are arranged from high to low, as slow polarization and diffusion processes gradually emerge, the EIS curve typically extends along the positive direction of the real part of the impedance. This allows for the construction of a monotonic constraint on the real part of the impedance. for: This constraint only penalizes the portion of the impedance where the real part decreases in the opposite direction, and does not force all samples to have the exact same slope.

[0040] Suppose the low-frequency spread tail contains the last K frequency points, with adjacent points having differential values. for: Diffusion Tail Stability Constraint for: This constraint limits drastic changes in the direction and step size of adjacent points at the low-frequency end, maintaining a stable extension trend of the diffusion tail. The physical constraint loss is: in, , and The weights of each physical constraint term.

[0041] Construct the total loss and train the model. Reconstruct the loss. Simultaneously calculate the real part of the impedance, the imaginary part of the impedance, and the impedance magnitude error: in, and They represent the first The real and imaginary parts of the measured complex impedance at each frequency point; and These represent the model at the 1st and 2nd. The real and imaginary parts of the impedance predicted at each frequency point; Indicates the first Measured complex impedance at each frequency point; The model represents the first time. The complex impedance is predicted at each frequency point.

[0042] KL divergence loss for: in, For the dimension of latent variables, Indicates the ordinal number of the latent variable dimension. Total loss function. for: in, and These represent the total weights of the KL divergence loss and the physical constraint loss, respectively. During the initial training phase, a KL weighted annealing strategy is used to... The value is gradually increased from a small value to avoid the latent variable regularization term prematurely inhibiting spectral reconstruction learning; when the validation set loss no longer decreases in several consecutive training rounds, an early stopping strategy is adopted to preserve the model parameters with the best generalization performance.

[0043] Step S4: Use Bayesian optimization to determine the model configuration. This includes setting the recurrent neural network type, hidden dimension, number of layers, latent variable dimension, true input probability, learning rate, batch size, and number of training epochs for both the encoder and decoder. , as well as , , Using parameters such as RMSE, MAPE, and coefficient of determination as candidate variables, and employing these as objective functions, the next set of candidate parameters is iteratively selected to obtain the optimal model. This avoids manual trial-and-error and improves the efficiency of the collaborative optimization of model structure and physical constraint weights.

[0044] Step S5 performs low-frequency prediction and full-frequency reconstruction. For the battery under test, only mid-to-high frequency EIS above a preset frequency threshold is collected, arranged and normalized in the same way as in the training phase, and then input into the model to generate a low-frequency EIS prediction sequence.

[0045] Step S6: Connect the mid-to-high frequency measured segment and the low-frequency predicted segment at the boundary frequency and perform inverse normalization to obtain the complete EIS. Further weighted fusion can be performed on a small number of overlapping frequency points near the boundary point to reduce discontinuities at the initial point.

[0046] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for rapid prediction of low-frequency EIS in lithium-ion batteries, characterized in that, include: Step S1: Obtain the full-frequency EIS of the lithium-ion battery under different states of charge, temperatures and operating conditions as training samples, where EIS is electrochemical impedance spectroscopy. The real and imaginary parts of the impedance at each frequency point of the training samples are arranged in descending order of frequency, and the full-frequency EIS is divided into a mid-to-high frequency input segment and a low-frequency target segment. Step S2: Construct a variational autoencoder with physical constraints; Step S3: Construct a total loss function consisting of reconstruction loss, KL divergence loss, and physical constraint loss, and train the physical constraint variational autoencoder using the training samples; Step S4: Use Bayesian optimization to determine the optimal physical constraint variational autoencoder configuration; Step S5: Acquire the mid-to-high frequency EIS of the lithium-ion battery under test, and input the mid-to-high frequency EIS of the lithium-ion battery under test into the optimally configured physical constraint variational autoencoder in step S4 to obtain the low frequency EIS prediction sequence of the lithium-ion battery under test. Step S6: The mid-to-high frequency EIS measured sequence of the lithium-ion battery under test and the low frequency EIS predicted sequence of the lithium-ion battery under test are spliced ​​together in frequency order to obtain the full frequency EIS of the lithium-ion battery under test.

2. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 1, characterized in that, The length of the mid-to-high frequency input segment is adjusted according to the requirements of measurement efficiency and prediction accuracy. Its proportion of the total number of full-frequency EIS frequency points is 40% to 80%, and the remaining frequency points constitute the low-frequency target segment.

3. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 1, characterized in that, The EIS includes steady-state EIS and dynamic EIS acquired under different charging states; wherein, the steady-state EIS is impedance data acquired after the battery has been left to stand and reached a relatively stable state, and the dynamic EIS is impedance data acquired directly during battery operation without being left to stand; the physical constraint variational autoencoder is suitable for low-frequency band prediction of steady-state EIS and dynamic EIS under different charging states.

4. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 1, characterized in that, The physical constraint variational autoencoder includes a recurrent neural network encoder and a conditional mapping module; The recurrent neural network encoder is used to extract frequency sequence features from the mid-to-high frequency input segment; the conditional mapping module is used to generate conditional vectors.

5. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 4, characterized in that, The physical constraint variational autoencoder further includes a latent variable generation module and a recurrent neural network autoregressive decoder; the latent variable generation module is used to obtain latent variables; the recurrent neural network autoregressive decoder generates low-frequency EIS point by point according to the condition vector, latent variables and impedance information of the previous frequency point.

6. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 5, characterized in that, The recurrent neural network autoregressive decoder initializes the decoder hidden state by concatenating the latent variables and the condition vector, and when generating the current low-frequency point, it uses the predicted impedance of the previous frequency point and the condition vector as input. During the training phase of the physical constraint variational autoencoder, the actual impedance of the previous frequency point is used to replace the predicted impedance with a preset probability, and the probability is gradually adjusted as the training progresses.

7. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 1, characterized in that, The physical constraint loss includes second-order difference smoothing constraint, impedance real part monotonicity constraint, and low-frequency diffusion tail stability constraint. The second-order difference smoothing constraint is used to limit the local curvature variation between adjacent low-frequency points; The monotonic constraint on the real part of the impedance is used to limit the non-physical decrease of the real part of the impedance as the frequency decreases.

8. The method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 7, characterized in that, The physical constraint loss also includes low-frequency diffusion tail stabilization constraint; The low-frequency diffusion tail stability constraint is used to limit the dispersion of the difference vector between adjacent frequency points at the low-frequency end.

9. A method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 7, characterized in that, The reconstruction loss includes the real part error of impedance, the imaginary part error of impedance, and the magnitude error of impedance; the KL divergence loss is used to make the posterior distribution of the latent variables output by the encoder close to the preset Gaussian prior distribution; the weight of the KL divergence loss in the total loss function is gradually increased from a small value in the early stage of training.

10. A method for rapid prediction of low-frequency EIS in lithium-ion batteries according to claim 1, characterized in that, Step S4 specifically includes: configuring the recurrent neural network type, hidden dimension, number of layers, latent variable dimension, true input probability, learning rate, batch size, and number of training epochs for the encoder and decoder. , Using α, β, and γ as candidate variables, and the validation set RMSE, MAPE, and coefficient of determination as objective functions, the next set of candidate parameters is iteratively selected to obtain the optimal physical constraint variational autoencoder.

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