Lithium-ion battery state of health estimation method based on physical information deep onet

Abstract

The application discloses a lithium ion battery health state estimation method based on physical information DeepONet. The method extracts multi-dimensional health characteristics from charge and discharge cycle data and constructs paired samples; a decoupled DeepONet architecture is constructed, a Branch-Trunk double branch structure is adopted to explicitly separate the working condition characteristics and the cycle index, and a sine activation function is adopted to ensure infinite differentiability; a physical discovery module is constructed, the partial derivatives of the prediction value with respect to time and characteristics are calculated through automatic differentiation, and the residual constraint and soft monotonicity constraint of ordinary differential equations are used to force the prediction trajectory to conform to the degradation dynamics; a two-stage training strategy is adopted, the optimal candidate model is selected through integrated initialization pre-training in the first stage, and physical regularization fine-tuning is performed on the dynamic weighted composite loss function in the second stage. The application alleviates the problems of feature confusion, gradient conflict and initialization sensitivity, and realizes high-precision and high-physical-consistency health state estimation.

Description

Technical Field

[0001] This invention belongs to the field of lithium-ion battery health management and intelligent state estimation, and particularly relates to a lithium-ion battery health state estimation method based on physical information DeepONet. Background Technology

[0002] Lithium-ion batteries, due to their high energy density and long cycle life, have become the primary power source for electric vehicles, portable electronic devices, and grid-scale energy storage systems. However, lithium-ion batteries inevitably undergo progressive aging during continuous charge-discharge cycles, manifested as capacity decay and increased impedance. State of Health (SOH), defined as the ratio of the current maximum discharge capacity to the rated nominal capacity, is a key indicator for quantifying degradation, maintenance scheduling, and predictive energy management. However, because batteries are complex electrochemical systems, their internal state cannot be directly measured. SOH must be inferred indirectly through observable signals such as voltage, current, and temperature, making SOH estimation a challenging indirect inference problem.

[0003] Accurate SOH estimation faces the following fundamental challenges: First, the underlying aging mechanism is highly nonlinear, involving cross-scale coupled processes such as solid electrolyte interfacial film growth, lithium deposition, and active material loss; second, acquiring high-quality degradation data is costly and time-consuming, resulting in a limited number of publicly available datasets; third, actual operating conditions vary significantly in terms of temperature, charge / discharge rate, and load curves, hindering model generalization; fourth, sensor noise, measurement drift, and capacity regeneration during rest periods introduce significant uncertainties; and fifth, purely data-driven models may produce physically unreasonable outputs, such as non-monotonic capacity trajectories, which are unacceptable in safety-critical applications.

[0004] Existing SOH estimation methods can be mainly categorized into model-based methods, data-driven methods, and hybrid methods. Model-based methods, such as the Extended Kalman Filter (EKF), while physically interpretable, suffer from high parameterization requirements and poor adaptability under heterogeneous conditions. Data-driven methods directly learn the mapping from measurable signals to SOH from recurrent data. These methods include fully connected networks (MLPs) for unified input processing, one-dimensional convolutional networks (CNNs) for local feature extraction, long short-term memory networks (LSTMs) for sequence degradation modeling, attention-enhanced bidirectional LSTMs (Attention-BiLSTMs) for adaptive feature weighting, and Transformer models that capture global dependencies through multi-head self-attention. While these methods demonstrate excellent accuracy within the distribution, they lack physical consistency guarantees and exhibit poor generalization ability under unseen conditions.

[0005] To bridge the gap between physical fidelity and data-driven flexibility, Physical Information Neural Networks (PINNs) embed physical constraints into neural network training. Wang et al. developed a PINN-based SOH framework, employing a paired-sample scheme for training on consecutive cyclic pairs, incorporating ODE residual constraints and monotonicity constraints. However, existing PINN methods still suffer from the following three key bottlenecks:

[0006] First, there's the issue of feature obfuscation at the architectural level. The existing PINN uses a holistic MLP architecture, concatenating condition-related sensor features (such as voltage and current statistics) and time-related cycle indices as a single input before feeding them into the network, thus obfuscating different physical mechanisms. Condition features describe the electrochemical state of the current cycle, while cycle indices reflect the time evolution of degradation; the two have fundamentally different physical meanings. Furthermore, piecewise linear activation functions like ReLU, widely used in MLPs, are non-differentiable at zeros. When training with physical information requires automatic differentiation to calculate higher-order time derivatives ∂û / ∂t in the ODE residuals, this leads to gradient discontinuities, compromising the reliability of the higher-order gradients required for ODE residual execution.

[0007] Second, there are gradient conflicts and initialization sensitivity issues at the optimization level. Simultaneously optimizing both data fidelity and physical residual objectives introduces gradient conflicts: data loss requires the network to fit the observations, while ODE residual loss requires the network output to satisfy differential equation constraints. The gradient directions of these two objectives may be opposite, leading to convergence instability. Random initialization further exacerbates training instability because physical information optimization based on automatic differentiation is highly sensitive to the quality of initial parameters—poor initialization can cause extremely noisy gradients in the ODE residual calculation during the early stages of training.

[0008] Third, there is a lack of systematic evaluation at the evaluation level. Most existing studies only validate on a single dataset and under limited operating conditions, lacking systematic evaluation across chemical systems (such as NCM, NCA), temperatures (25°C to 45°C), and cyclic protocols (constant current, random discharge, real driving conditions, etc.). Summary of the Invention

[0009] To address the aforementioned problems of existing methods, this invention proposes a lithium-ion battery health state estimation method based on physical information DeepONet and integrated pre-training and dynamic loss weighting, comprising the following steps:

[0010] A method for estimating the health status of lithium-ion batteries based on physical information DeepONet includes the following steps: Step 1: Extract health features from lithium-ion battery charge-discharge cycle data and construct paired samples containing operating condition features and cycle indices; Step 2: Construct a decoupled DeepONet architecture, which includes a branch network and a backbone network. The branch network is used to encode operating condition features, and the backbone network is used to encode cycle indices. The outputs of the two are combined through an inner product and then passed through an output network to obtain the predicted health status value; Step 3: Construct a physical discovery module, which calculates the partial derivatives of the predicted health status value with respect to time and features through automatic differentiation, concatenates them into an augmented state vector, and inputs it into a latent dynamics network. The predicted trajectory is forced to conform to degradation dynamics through ordinary differential equation residual constraints and monotonicity constraints; Step 4: Adopt a two-stage training strategy. In the first stage, the optimal candidate model is selected through integrated initialization pre-training. In the second stage, physical regularization fine-tuning is performed using a dynamically weighted composite loss function, and the trained model is output for health status estimation.

[0011] Furthermore, the health features mentioned in step 1 include 16-dimensional features extracted from the constant current charging stage and the constant voltage charging stage, specifically including: voltage domain features composed of voltage mean, voltage standard deviation, voltage kurtosis, voltage skewness, voltage slope, and voltage entropy; current domain features composed of current mean, current standard deviation, current kurtosis, current skewness, current slope, and current entropy; capacity features composed of constant current charging capacity and constant voltage charging capacity; and time features composed of constant current charging time and constant voltage charging time. The paired samples are constructed by pairing the vector composed of the 16-dimensional features of each cycle and the cycle index with the corresponding vector of the adjacent cycle.

[0012] Furthermore, the decoupled DeepONet architecture described in step 2 is as follows: the branch network acts as a working condition encoder, mapping the normalized working condition feature vectors to a time-invariant latent representation. Its network structure includes an input layer, a hidden layer, and an output layer. The hidden layer uses a sinusoidal activation function and a Dropout layer. The backbone network acts as a time basis function encoder, mapping the scalar cyclic index to basis coefficients. Its network structure is symmetrical to the branch network. The outputs of the two sub-networks are multiplied element-wise and then summed along the latent dimension to obtain a scalar interaction value, which is then passed through the output network to obtain the health status prediction value. This achieves explicit decoupling of working condition features and time information at the architectural level.

[0013] Furthermore, both the branch network and the backbone network use a sinusoidal activation function instead of the ReLU activation function, making the network output infinitely differentiable. This ensures the numerical stability when calculating higher-order time derivatives through automatic differentiation and avoids the gradient discontinuity problem caused by the non-differentiability of the ReLU activation function at zero.

[0014] Furthermore, the physical discovery module in step 3 specifically includes: calculating the partial derivatives of the health state prediction value with respect to the cyclic index and with respect to the operating condition features through automatic differentiation; concatenating the original input operating condition features, cyclic index, health state prediction value, partial derivatives of the prediction value with respect to the operating condition features, and partial derivatives of the prediction value with respect to the cyclic index into an augmented state vector; inputting the augmented state vector into a latent dynamics network using a sinusoidal activation function to obtain the estimated degradation rate; defining the difference between the partial derivative of the prediction value with respect to the cyclic index and the output of the latent dynamics network as the ordinary differential equation residual, and forcing the predicted trajectory to conform to the degradation dynamics by minimizing this residual.

[0015] Furthermore, the monotonicity constraint adopts a soft constraint form, and the monotonicity constraint loss is calculated by paired samples. When the model predicts a later cycle health state that is higher than an earlier cycle health state, but the actual health state is declining, a positive penalty term is generated. When the predicted change direction is consistent with the actual change direction, the penalty is zero, thereby maintaining the ability to model capacity regeneration phenomena while maintaining physical rationality.

[0016] Furthermore, the first stage of integrated initialization pre-training in step 4 specifically includes: constructing multiple replicas of the prediction network with the same structure but different initial parameters, each replica being initialized independently and randomly; performing independent pre-training on each replica using only data loss, without introducing physical constraints; after training, selecting the candidate model with the smallest validation error, using its parameters as the initialization of the second stage prediction network, and simultaneously copying the optimal model as a fixed prior model for calculating the second stage prior regularization loss.

[0017] Furthermore, in step 4, the dynamically weighted composite loss function of the second stage includes data loss, ordinary differential equation residual loss, monotonicity constraint loss, and prior regularization loss. The data loss is used to ensure the model's fitting accuracy to the observed data. The ordinary differential equation residual loss is used to force the predicted trajectory to conform to the degradation dynamics. The monotonicity constraint loss is used to prevent physically inconsistent capacity rebound. The prior regularization loss is used to calculate the error between the current model prediction and the fixed prior model prediction in the first stage, preventing the model from deviating from the data-driven optimal solution during the transition period introduced by physical constraints.

[0018] Furthermore, the dynamic weighting employs a complementary scheduling strategy: the weights of the ordinary differential equation residual loss and the monotonicity constraint loss gradually increase in the first half of the training process, enabling the model to first establish a stable data-driven baseline and then gradually accept the guidance of physical constraints; the weights of the prior regularization loss gradually decay to zero in the early stage of training, anchoring the model near the pre-training state to maintain stability in the early stage of training, and gradually releasing them as physical constraints gradually assume the role of regularization; the weights of the four losses are dynamically adjusted with the progress of normalized training, forming a progressive course learning paradigm from data-driven fitting to physical constraint refinement.

[0019] Furthermore, in step 4, the second stage uses two independent optimizers to update the parameters of the prediction network and the dynamics network respectively. The prediction network uses a cosine annealing learning rate scheduling strategy with preheating to balance multiple competing losses, while the dynamics network uses a fixed learning rate to continuously track the degradation rate field defined by the prediction network.

[0020] The beneficial effects of this invention are as follows:

[0021] 1. A decoupled DeepONet architecture is proposed. A branch-backbone dual-branch structure explicitly decouples the 16-dimensional working condition sensor features from the cyclic index time coordinates. This ensures that the latent representation encoded by the branch network is naturally time-invariant electrochemical features, while the basis coefficients encoded by the backbone network focus on the temporal evolution of degradation dynamics, avoiding the interference between working condition information and time information in traditional monolithic architectures. Furthermore, both the branch and backbone networks use sinusoidal activation functions instead of ReLU, guaranteeing the infinite differentiability of the network output and ensuring the numerical stability of the higher-order time derivatives calculated through automatic differentiation during the physical information training phase. In addition, the soft monotonicity constraint, while tolerating transient capacity regeneration supported by data, effectively prevents physically inconsistent SOH bounce, improving the physical consistency of SOH estimation.

[0022] 2. A two-stage training strategy and dynamic course weighting mechanism for integrated initialization were designed to solve two major optimization challenges in standard PINN: gradient conflict and initialization sensitivity. The first stage initializes multiple candidate prediction networks in parallel, pre-trains independently using only data loss, and then selects the model with the smallest validation error for the second stage initialization. This effectively avoids the ODE residual noise amplification problem that may be caused by single random initialization. The second stage employs a dynamic loss weighting strategy: the physical constraint weights gradually increase from small values ​​to the target value as training progresses, while the prior regularization weights synchronously decay from high values ​​to zero. This achieves a smooth transition from data-driven fitting to physical constraint refinement. During training, the PDE loss gradually decreases from a small value to a target value. The magnitude steadily decreased The magnitude was not significant and no serious oscillations occurred.

[0023] 3. Fully validated on two heterogeneous benchmark datasets (University A and University B), PI-DeepONet reduces RMSE by up to 29.8% compared to the best data-driven baseline and by up to 40.6% compared to the standard PINN.

[0024] 4. Physical constraints not only improve average accuracy but also constrain the maximum prediction error, especially achieving significantly tighter tracking accuracy in the late-stage degradation phase when SOH is below 0.85. Attached Figure Description

[0025] Figure 1 This is a flowchart of the overall PI-DeepONet framework of the present invention;

[0026] Figure 2 This is a schematic diagram of the DeepONet architecture for decoupled physical information according to the present invention, which includes a branch network, a backbone network, an inner product operation, and an output network.

[0027] Figure 3 This is a schematic diagram of the original discharge capacity degradation curves of all batteries in three batches of a dataset from a certain University B. In the figure, batch 1 is NCA battery, batch 2 is NCM battery, and batch 3 is NCM+NCA hybrid battery.

[0028] Figure 4 This is a schematic diagram showing the evolution trend of 16 normalized health characteristics of NCM batteries from batch 2 of a certain university B under three temperatures (25°C, 35°C, and 45°C) with aging cycles.

[0029] Figure 5a A bar chart showing the RMSE comparison of each method on the first batch of batteries in a dataset from University A.

[0030] Figure 5b A bar chart showing the RMSE comparison of each method on the second batch of batteries in a dataset from University A.

[0031] Figure 5c A bar chart showing the RMSE comparison of each method on the third batch of batteries in a dataset from University A.

[0032] Figure 5d A bar chart showing the RMSE comparison of each method on the fourth batch of batteries in a dataset from University A.

[0033] Figure 5e A bar chart showing the RMSE comparison of each method on the fifth batch of batteries in a dataset from University A.

[0034] Figure 5f A bar chart showing the RMSE comparison of each method on the sixth batch of batteries in a dataset from University A.

[0035] Figure 6aThis is a scatter plot of the predicted SOH and the actual SOH of the first batch of batteries in a dataset from University B. The dashed line is the ideal y=x reference line.

[0036] Figure 6b This is a scatter plot of the predicted SOH and the actual SOH of the second batch of batteries in a dataset from University B. The dashed line is the ideal y=x reference line.

[0037] Figure 6c This is a scatter plot of the predicted SOH and the actual SOH of the third batch of batteries in a dataset from University B. The dashed line is the ideal y=x reference line.

[0038] Figure 7 The cloud and rain map shows the prediction error distribution of all methods on the first batch of batteries from a certain University A.

[0039] Figure 8a This is a schematic diagram of the loss convergence curve during Phase 2 training.

[0040] Figure 8b This is a dynamic weight scheduling graph during the Phase 2 training process. Detailed Implementation

[0041] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0042] The lithium-ion battery health state estimation method based on DeepONet physical information of this invention first extracts multi-dimensional health features from lithium-ion battery charge-discharge cycle data, and constructs a system as follows: Figure 1 The decoupled DeepONet architecture (PI-DeepONet) shown separates condition features from the time coordinate. Then, a neural ODE physics discovery module learns the latent degradation dynamics, and monotonicity constraints are applied to ensure physical consistency. Finally, stable convergence is achieved through a two-stage training strategy integrating initialization and dynamic loss weighting.

[0043] like Figure 2 As shown, the method of the present invention specifically includes the following steps:

[0044] Step 1: Data Acquisition and Feature Extraction: Feature extraction and preprocessing are performed on the charge and discharge cycle data of lithium-ion batteries. 16-dimensional health features are extracted from the constant current charging stage and the constant voltage charging stage. After smoothing filtering, outlier removal and standardization, normalized feature vectors are obtained, and paired samples are constructed according to adjacent cycles.

[0045] The specific process of data acquisition and feature extraction is as follows:

[0046] Acquire charge-discharge cycle data of lithium-ion batteries. Sixteen health features are extracted from the constant-current (CC) and constant-voltage (CV) charging phases of each cycle. Relying solely on voltage and current measurements during charging, the method requires no adjustments for specific chemical systems or protocols, ensuring its versatility across different battery types. The features are organized into four groups:

[0047] 1) Voltage domain characteristics (CC stage): six characteristics in total: voltage mean, voltage standard deviation, voltage kurtosis, voltage skewness, voltage slope, and voltage entropy;

[0048] 2) Current domain characteristics (CV stage): six characteristics in total: current mean, current standard deviation, current kurtosis, current skewness, current slope, and current entropy.

[0049] 3) Capacity characteristics: There are two constant current charging capacity (CCQ) and constant voltage charging capacity (CVQ);

[0050] 4) Time characteristics: There are two time characteristics: constant current charging time and constant voltage charging time.

[0051] Among these features, the CC-Q and CV charging times exhibit a monotonic trend with aging, directly reflecting impedance growth and serving as strong indicators of state of equilibrium (SOH). In contrast, features such as voltage kurtosis and current skewness exhibit non-monotonic behavior, capturing complementary information about degradation mode transitions. These diverse characteristics collectively provide rich degradation characterizations for subsequent models.

[0052] The original capacity curve was smoothed using a Savitzky-Golay filter to suppress sensor noise while preserving the degradation trend. This is crucial for the physical information training stage, as this stage requires automatic differentiation to calculate the time derivative ∂û / ∂t, and noisy trajectories can lead to unstable residual loss in the differential equation. The SOH label is defined as SOH = Q_current / Q_nominal × 100, where Q_current is the filtered discharge capacity and Q_nominal is the nominal capacity. Outlier removal was performed on the extracted features using a three-standard-deviation rule to eliminate outliers caused by sensor failure or extreme operating conditions. The retained features were then z-score standardized on the training set. , This represents the original value of the j-th feature. This represents the mean of the j-th feature on the training set. Let $\mathbf{j}$ represent the standard deviation of the $j$-th feature on the training set, where $\mathbf{j}$ is the standard deviation of the $j$-th feature on the training set. and Computation is performed only on the training set to avoid leaking information from the test set.

[0053] For each battery's cyclic data, the 17-dimensional vector (16-dimensional features + cycle index) of the i-th cycle is paired with the vector of the (i+1)-th cycle to form training samples (x1, y1) and (x2, y2). This paired sample scheme allows subsequent monotonicity constraints to utilize the temporal relationship between adjacent cycles, encoding prior knowledge of capacity reduction without relying on explicit physical equations. The dataset partitioning employs a two-layer protocol based on individual batteries: at the battery level, batteries under various operating conditions are divided into training and testing groups, ensuring that the test set consists of battery individuals never seen during the training phase, reflecting the scenario of predicting new batteries in actual deployment; the training group is randomly divided into training and validation subsets in an 8:2 ratio.

[0054] Step 2: Construct a decoupled DeepONet architecture for SOH prediction. The DeepONet architecture consists of three parts: a branch network (work condition encoder), a backbone network (time basis function encoder), and an output network. The branch network encodes 16-dimensional work condition features, and the backbone network encodes the cyclic index. The two are combined through an inner product and then passed through the output network to obtain the predicted value. Both the Branch Net and the Trunk Net use a sinusoidal activation function to ensure infinite differentiability.

[0055] The specific construction process of the decoupled DeepONet architecture is as follows:

[0056] Existing PINN methods generally employ a holistic MLP architecture, concatenating operating condition features and cycle indices as unified inputs before feeding them into the network. This early fusion strategy obscures the cumulative effects of intrinsic electrochemical kinetics and operating time, limiting generalization ability when switching between different operating condition protocols. To address this issue, this invention constructs a decoupled architecture based on operator learning, explicitly separating degradation-related operating conditions from temporal dynamics through a dual-branch structure.

[0057] Branch network (work condition encoder): maps the 16-dimensional normalized feature vector x to a latent representation. The network structure is as follows: the first layer is a linear layer (16-dimensional input, p-dimensional output) followed by a sinusoidal activation function; the middle hidden layers are linear layers followed by sinusoidal activation functions and Dropout layers; and the last layer is a linear layer. Because the input... Not containing circular indexes The result represents Naturally time-invariant, it encodes the electrochemical characteristics under specific operating stress factors—such as the degradation modes corresponding to specific temperature, charge / discharge rate, and load curve combinations—enabling the model to distinguish the differentiated effects of different operating conditions on battery aging.

[0058] Backbone network (time basis function encoder): maps scalar cyclic index t to basis coefficients. The network structure is symmetrical with respect to the branch network, and the input dimension is 1. The basis coefficients encode the temporal evolution of the degradation dynamics, independent of specific operating conditions. This separation allows the model to adjust the operating condition representation only through the branch network when the operating condition characteristics change (such as switching from a constant current charging protocol to a random discharge protocol), while the basic pattern of temporal evolution is kept stable by the Trunk Net, thereby enhancing the generalization ability across operating conditions.

[0059] The outputs of the two subnetworks are multiplied element-wise and summed along the latent dimension to obtain a scalar interaction value, which is then passed through the output network. The predicted SOH value was obtained:

[0060]

[0061] in These are the trainable parameters for the branch network, backbone network, and output network, respectively. Output network (·) represents a linear-sinusoidal activated linear projection block. The first potential representation of the branch network output Each component represents an electrochemical characteristic related to the coding conditions. The first of the basic coefficients of the backbone network output Each component encodes the temporal evolution of degradation dynamics. Unlike the standard DeepONet which directly uses the inner product as the output, this invention introduces controllable nonlinearity through this compact nonlinear post-processing network to better approximate the complex fluctuations superimposed on the long-term degradation trend.

[0062] Both the branch network and the backbone network use sinusoidal activation functions. It replaces the traditional ReLU activation function. This represents the output value of the linear layer in the neural network, i.e., the input of the activation function. The sine function was chosen based on two considerations: First, the SOH trajectory exhibits nonlinear fluctuations superimposed on a long-term degradation trend, and the sine function is naturally suitable for representing multi-scale oscillation modes; second, the sine function has infinite differentiability (…). Its derivative String-like functions guarantee the existence and numerical stability of derivatives of any order. ReLU, however, is not differentiable at zeros, and its higher-order time derivatives are calculated automatically during physical information training. This can lead to gradient discontinuities, compromising the reliability of the ODE residual constraint. Under the sinusoidal activation function, the time derivative of the predicted SOH with respect to the cyclic index can be analytically expressed as:

[0063]

[0064] This smoothness significantly enhances compatibility with residual-based physical constraints in subsequent step 3.

[0065] The actual network parameters are configured as follows: both the branch network and the backbone network contain 3 hidden layers, each with 48 units, and a Dropout rate of 0.2. The Dropout layer randomly masks some neurons during training to prevent overfitting. All linear layer parameters are initialized using the Xavier normal distribution, and the bias is initialized to zero. This initialization scheme maintains the consistency of the variance of activation values ​​in each layer, which is beneficial for gradient flow in deep networks.

[0066] Step 3: Construct a physics discovery module. This module automatically differentiates the partial derivatives of the predicted values ​​with respect to time and features, concatenates them into an augmented state vector, and inputs it into the latent dynamics network. The predicted trajectory is then constrained to conform to degenerate dynamics through ODE residuals. Simultaneously, monotonic constraints are applied to prevent physically inconsistent capacity recovery.

[0067] The construction process of the physics discovery module is as follows:

[0068] Battery degradation is continuously driven by electrochemical processes such as solid electrolyte interfacial film growth, lithium deposition, and loss of active materials. The degradation trajectory follows a general form of ordinary differential equation: ,in For true SOH, The degradation dynamics are unknown. However, due to the highly nonlinear degradation mechanism and the involvement of multi-scale coupled processes, it remains to be seen how this will affect the degradation process. Specifying the parsing form is extremely difficult. This invention does not specify... Instead of using a specific form, it uses a learnable latent dynamic network. The method autonomously discovers degradation patterns from data, balancing the guiding role of physical constraints with the flexibility of data-driven approaches.

[0069] Specifically, the 17-dimensional input vector is divided into the first 16 feature dimensions. and last cyclic index The concatenated data is then fed into DeepONet to obtain the predicted values. Calculate separately using automatic differentiation. right partial derivatives and against partial derivatives The computation graph is preserved during the operation to support backpropagation of the second-order gradient. The original input, predicted values, and partial derivatives are concatenated into an augmented state vector.

[0070]

[0071] Among them, 16 dimensions Capture the sensitivity of the predicted value to various input features, such as the degree of response to temperature changes or charge / discharge rate changes; 1D Capturing instantaneous degradation rates reflects the rate of battery capacity decay at the current cycle. Adding this gradient information to the augmented state vector provides the dynamical network with rich local information about the solution manifold, enabling it to perceive local trends in predicted values ​​across the feature space and time dimension, thus more accurately approximating the unknown degradation control equations.

[0072] Potential dynamical network F: This is a multilayer perceptron using a sinusoidal activation function, with 3 hidden layers, 60 units per layer, and a dropout rate of 0.2. The ODE residual is defined as...

[0073]

[0074] Physical information training is forced on all observation training cycles. That is, the predicted rate of change of SOH is required. Degradation rate estimated by dynamic network Maintaining consistency ensures that the predicted SOH trajectory follows the autonomously discovered degradation dynamics while satisfying data fitting requirements, thus avoiding physically unreasonable prediction behavior.

[0075] In addition to the ODE residual constraint, this invention also applies a soft monotonicity constraint to further ensure physical consistency. Battery capacity generally decreases with increasing cycle number, but transient capacity regeneration may occur during rest periods. To accommodate this characteristic, the monotonicity constraint loss is designed as follows:

[0076]

[0077] in This represents the model-predicted SOH value for a later cycle (the (i+1)th cycle) in the paired samples. This represents the model-predicted SOH value for an earlier cycle (the i-th cycle) in the paired samples. This indicates the actual SOH value corresponding to an earlier cycle. This indicates the actual SOH value corresponding to a later cycle. This represents the total number of paired samples in the training set, i.e., the number of sample pairs involved in the monotonicity constraint loss calculation.

[0078] A positive penalty term is generated when the model predicts an increase in SOH while the actual value decreases; the penalty is zero when the two changes in the same direction. The brilliance of this soft constraint lies in the fact that it does not require SOH to be strictly monotonically decreasing, but only applies a penalty when the model's predicted trend contradicts the actual trend, thus preserving the ability to model capacity regeneration phenomena while maintaining physical rationality.

[0079] Step 4: Adopt a two-stage training strategy: The first stage integrates initialization pre-training to select the optimal candidate model, and the second stage uses a dynamically weighted composite loss function for physical regularization fine-tuning. Finally, the trained model is output for SOH estimation.

[0080] The two-stage training strategy is as follows:

[0081] Training directly from random initialization under physical constraints faces two major challenges: First, the ODE residuals, calculated through automatic differentiation, are highly sensitive to the quality of initial parameters; poor initialization can cause extremely noisy gradients in the ODE residuals during the early stages of training. Second, the gradient directions of data loss and physical loss may be opposite, and optimizing both simultaneously can easily lead to convergence oscillations. To address these challenges, this invention designs a two-stage training strategy integrating initialization.

[0082] Phase 1 – Integrated Initialization Pre-training: M=3 identical prediction network replicas with different initial parameters are constructed. The weights of each replica are independently and randomly initialized using a Xavier normal distribution. By training multiple candidate models in parallel instead of relying on a single initialization, the risks associated with initialization randomness are effectively mitigated. Each candidate model is configured with an independent optimizer and a cosine annealing learning rate scheduler with linear preheating (15 rounds of preheating). Rise to Then, according to cosine decay to Each candidate model is trained for a maximum of 100 epochs, with an early termination period of 20 epochs. Only data loss is used in this phase.

[0083]

[0084] Training, without introducing any physical constraints, aims to allow the model to first find a reasonable starting point for fitting within the data space. After training, the candidate model with the smallest validation error is selected.

[0085]

[0086] in This represents the optimal parameters for the m-th candidate model when it achieves the lowest validation error (MSE) during the pre-training phase.

[0087] Use its parameters as the initialization for the second stage. At the same time, make a copy of this optimal model as a fixed prior model (the parameters will not be updated) for the calculation of the prior regularization loss in the second stage.

[0088] The second stage—physical regularization fine-tuning—starts from the high-quality initialization selected in the first stage, jointly optimizing the prediction network and the dynamics network. Minimizing the composite objective function:

[0089]

[0090] Where t=e / E∈[0,1] is the normalized training progress. The mean squared error of the prediction for paired samples ensures the model's fitting accuracy to the observed data;

[0091] The ODE residual loss The same calculation is performed under the paired sample scheme:

[0092]

[0093] in The ODE residual defined in step 3. This loss forces the predicted trajectory to conform to degradation dynamics.

[0094] The monotonic constraint loss Calculate as defined in step 3 to prevent physically inconsistent capacity rebound.

[0095] The prior regularization loss Calculate the mean squared error between the current model prediction and the prediction of the first-stage fixed prior model:

[0096]

[0097] in This is the output of a fixed copy of the optimal pre-trained model from the first stage; the parameters of this fixed copy are not updated during the computation. This loss prevents the model from deviating from the data-driven optimal solution during the transition period introduced by physical constraints.

[0098] The weights of the four loss terms employ a dynamic scheduling strategy, which is the core mechanism for resolving gradient conflicts in this invention. PDE Loss Weights It grows linearly from 0.1 to 0.5 ( Monotonicity constraint weights It increased linearly from 0.05 to 0.2 ( Both methods are gradually enhanced in the first half of training, allowing the model to first establish a stable data-driven baseline before gradually accepting the guidance of physical constraints. This avoids the noise gradients of physical loss in the early stages of training damaging the still-unstable network parameters. Prior regularized weights. It decays linearly from 1.0 to 0.0 ( In the early stages of training, the model is anchored near the pre-trained state to maintain stability. As physical constraints gradually assume a regularization role, the model is gradually released and eventually exits completely. This complementary scheduling forms a progressive learning paradigm from data-driven fitting to physical constraint refinement.

[0099] The second stage uses two independent optimizers to update the prediction network parameters separately. and dynamic network parameters The prediction network employs a cosine annealing learning rate schedule with preheating (30 rounds of preheating). Rise to After decaying to This requires fine-tuning of the learning rate to balance the four competing losses. Dynamical networks use a fixed learning rate. Because it only needs to track the degradation rate field defined by the prediction network, using a stable fixed learning rate is more conducive to continuously tracking the evolution of the prediction network. The maximum training duration is 200 epochs, with an early termination period of 20 epochs, and a batch size of 256. The entire two-stage pipeline is executed independently 10 times, and the average result is reported.

[0100] Example:

[0101] 1. Dataset

[0102] Two publicly available lithium-ion battery aging datasets were selected, totaling 185 batteries and more than 80,762 cycle-level samples.

[0103] like Figures 5a-5f As shown, a dataset from University A contains 55 NCM batteries (nominal capacity 2000mAh, voltage range 2.5-4.2V) across 6 batches. These batteries were cycled under six protocols: 2C constant current, 3C constant current, R2.5 random discharge, R3 random discharge, RW real driving conditions, and satellite power simulation, with cycle lives ranging from 121 to 942 cycles. Batteries numbered with 4 or 8 under each condition were used as the test set.

[0104] like Figure 3 Figure 4 As shown, a dataset from University B covers three chemical systems: Batch 1 consists of 66 NCA batteries (3500mAh), Batch 2 consists of 55 NCM batteries (3500mAh), all aged at 25°C, 35°C, and 45°C; Batch 3 consists of 9 NCM+NCA hybrid batteries (2500mAh), aged only at 25°C. Cycle life ranges from 32 to 1050 cycles.

[0105] 2. Baseline Method

[0106] Compared to seven baseline methods: MLP (3 layers, 60 units), CNN (5 residual blocks), LSTM (2 layers, 64 units), Attention-BiLSTM (2 layers, 64 units, bidirectional LSTM + attention), Transformer (3 layers, encoder, d_model=64), PINN (MLP architecture + neural ODE + static weights), and DeepONet-only (DeepONet with only data loss). All methods were trained under the same data partitioning and evaluation protocol.

[0107] 3. Experimental Environment

[0108] The experiment used an NVIDIA GeForce RTX 4050 Laptop GPU and the PyTorch deep learning framework. The models were implemented in Python. Since CPU-based model training is relatively slow, GPU-based training of the network framework was used to accelerate computation. In the pre-training phase (Phase 1), the number of ensemble candidate models was 3, with each candidate model trained for a maximum of 100 epochs. In the formal training phase (Phase 2), the maximum number of training epochs was 200, with a batch size of 256. The loss functions included mean squared error loss, ODE residual loss, monotonicity constraint loss, and prior regularization loss. To account for randomness, the entire two-stage training pipeline was executed independently 10 times, and the results were reported as the mean. The evaluation metrics used were root mean square error (RMSE) and mean absolute percentage error (MAPE).

[0109] 4. Performance Verification

[0110] Two metrics, RMSE and MAPE, were used. On a dataset from University A (batch 1), PI-DeepONet achieved an RMSE of 0.0054, a 29.8% reduction compared to Attention-BiLSTM (0.0077) and a 40.6% reduction compared to PINN (0.0091). On batches 4 and 6, the reductions were 10.4% and 9.3%, respectively. On a dataset from University B, batch 1 achieved RMSE of 0.0149, batch 2 achieved RMSE of 0.0126, and batch 3 achieved RMSE of 0.0060.

[0111] Ablation experiments show that transitioning from MLP to the DeepONet architecture reduces RMSE by 25.3%, and further reduces it after adding pre-training. The complete framework (integrating pre-training and dynamic weighting) reduces RMSE by an additional 11.5% in batch 1.

[0112] like Figures 6a-6c As shown in the figure, the scatter plots of predicted SOH and actual SOH for three batches of a dataset from University B are as follows. Figure 7 The image shown is a cloud and rain map depicting the prediction error distribution of all methods on the first batch of batteries from University A. Figures 8a-8b As shown, the data loss monotonically decreases during training, and the PDE loss increases from... The magnitude dropped to The magnitude of the results, without severe oscillations, verifies that the dynamic weighting strategy successfully mitigated gradient conflicts.

[0113] Finally, it should be noted that the purpose of disclosing the embodiments is to help further understand the present invention. However, those skilled in the art will understand that various substitutions and modifications are possible without departing from the spirit and scope of the present invention and the appended claims. Therefore, the present invention should not be limited to the content disclosed in the embodiments, and the scope of protection of the present invention is defined by the claims.

Claims

1. A method for estimating the state of health of lithium-ion batteries based on physical information DeepONet, characterized in that, Includes the following steps: Step 1: Extract health features from lithium-ion battery charge-discharge cycle data and construct paired samples containing operating condition features and cycle indexes; Step 2: Construct a decoupled DeepONet architecture, which includes a branch network and a backbone network. The branch network encodes working condition features, and the backbone network encodes cyclic indices. The outputs of the two are combined through an inner product and then passed through an output network to obtain the predicted health status value. Specifically, the decoupled DeepONet architecture is as follows: the branch network acts as a working condition encoder, mapping the normalized working condition feature vectors to time-invariant latent representations. Its network structure includes an input layer, hidden layers, and an output layer. The hidden layers employ a sinusoidal activation function and a Dropout layer. The backbone network acts as a time basis function encoder, mapping scalar cyclic indices to basis coefficients. Its network structure is symmetrical to that of the branch network. The outputs of each sub-network are multiplied element-wise and then summed along the latent dimension to obtain a scalar interaction value, which is then passed through the output network to obtain the health status prediction value, thereby achieving explicit decoupling of working condition features and time information at the architectural level; Step 3: Construct a physical discovery module, which calculates the partial derivatives of the health status prediction value with respect to time and features through automatic differentiation, splices them into an augmented state vector and inputs it into the latent dynamics network, and forces the predicted trajectory to conform to the degradation dynamics through ordinary differential equation residual constraints and monotonicity constraints; Step 4: Adopt a two-stage training strategy. In the first stage, the optimal candidate model is selected through integrated initialization pre-training. In the second stage, physical regularization fine-tuning is performed using a dynamically weighted composite loss function, and the trained model is output for health status estimation.

2. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 1, characterized in that, The health features mentioned in step 1 include 16-dimensional features extracted from the constant current charging stage and the constant voltage charging stage. Specifically, these include: voltage domain features consisting of voltage mean, voltage standard deviation, voltage kurtosis, voltage skewness, voltage slope, and voltage entropy; current domain features consisting of current mean, current standard deviation, current kurtosis, current skewness, current slope, and current entropy; capacity features consisting of constant current charging capacity and constant voltage charging capacity; and time features consisting of constant current charging time and constant voltage charging time. The paired samples are constructed by pairing the vector formed by the 16-dimensional features of each cycle and the cycle index with the corresponding vector of the adjacent cycle.

3. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 2, characterized in that, Both the branch network and the backbone network use a sinusoidal activation function instead of the ReLU activation function, making the network output infinitely differentiable. This ensures the numerical stability when calculating higher-order time derivatives through automatic differentiation and avoids the gradient discontinuity problem caused by the non-differentiability of the ReLU activation function at zero.

4. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 1, characterized in that, The physical discovery module in step 3 specifically includes: calculating the partial derivatives of the health status prediction with respect to the cyclic index and with respect to the operating condition features through automatic differentiation; concatenating the original input operating condition features, cyclic index, health status prediction, partial derivatives of the prediction with respect to the operating condition features, and partial derivatives of the prediction with respect to the cyclic index into an augmented state vector; inputting this augmented state vector into a latent dynamics network using a sinusoidal activation function to obtain the estimated degradation rate; defining the difference between the partial derivative of the prediction with respect to the cyclic index and the output of the latent dynamics network as the ordinary differential equation residual, and forcing the predicted trajectory to conform to the degradation dynamics by minimizing this residual.

5. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 4, characterized in that, The monotonicity constraint adopts a soft constraint form, and the monotonicity constraint loss is calculated by paired samples. When the model predicts a later cycle health state that is higher than an earlier cycle health state, but the actual health state is declining, a positive penalty term is generated. When the predicted change direction is consistent with the actual change direction, the penalty is zero, thereby maintaining the ability to model capacity regeneration phenomena while maintaining physical rationality.

6. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 1, characterized in that, Step 4, the first stage of integrated initialization pre-training, specifically includes: constructing multiple replicas of the prediction network with the same structure but different initial parameters, each replica being initialized independently and randomly; performing independent pre-training on each replica using only data loss, without introducing physical constraints; after training, selecting the candidate model with the smallest validation error, using its parameters as the initialization of the second stage prediction network, and simultaneously copying the candidate model as a fixed prior model for calculating the second stage prior regularization loss.

7. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 1, characterized in that, In step 4, the dynamically weighted composite loss function of the second stage includes data loss, ordinary differential equation residual loss, monotonicity constraint loss, and prior regularization loss. The data loss is used to ensure the model's fitting accuracy to the observed data. The ordinary differential equation residual loss is used to force the predicted trajectory to conform to the degradation dynamics. The monotonicity constraint loss is used to prevent physically inconsistent capacity rebound. The prior regularization loss is used to calculate the error between the current model prediction and the fixed prior model prediction in the first stage, preventing the model from deviating from the data-driven optimal solution during the transition period introduced by physical constraints.

8. The lithium-ion battery health state estimation method based on physical information DeepONet according to claim 7, characterized in that, The dynamic weighting employs a complementary scheduling strategy: the weights of the ordinary differential equation residual loss and the monotonicity constraint loss gradually increase in the first half of training, enabling the model to first establish a stable data-driven baseline and then gradually accept the guidance of physical constraints; the weights of the prior regularization loss gradually decay to zero in the early stages of training, anchoring the model near the pre-training state to maintain stability in the early stages of training, and gradually releasing them as physical constraints gradually assume the role of regularization; the weights of the four losses are dynamically adjusted with the progress of normalized training, forming a progressive learning paradigm from data-driven fitting to physical constraint refinement.

9. The lithium-ion battery health state estimation method based on DeepONet physical information according to claim 1, characterized in that, In step 4, the second stage uses two independent optimizers to update the parameters of the prediction network and the dynamics network respectively. The prediction network uses a cosine annealing learning rate scheduling strategy with preheating to balance multiple competing losses, while the dynamics network uses a fixed learning rate to continuously track the degradation rate field defined by the prediction network.

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