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Integrating Physics-Based Modeling with Machine Learning for Lithium-Ion Batteries
Hao Tu, Scott Moura, Yebin Wang, Huazhen Fang
TL;DR
Accurate and computationally efficient LiB models remain difficult because physics-based models trade fidelity against cost, while existing hybrid models can lack physically meaningful mappings. This paper introduces two state-informed physics-ML frameworks, integrates electrochemical and equivalent-circuit models with FNNs, and reports accurate voltage prediction across broad C-rates and throughout cycle life with aging awareness.
Problem
LiB management requires accurate and efficient dynamic models, but physics-based models face computation or accuracy limits and existing hybrid models can learn physically unmeaningful mappings.
Method
The paper develops HYBRID-1 and HYBRID-2, which integrate physical LiB models with FNNs while feeding the FNN physical-model state information.
Results
The resulting electrochemical- and ECM-based hybrid models demonstrate high voltage predictive accuracy across broad C-rate ranges, with an aging-aware model predicting throughout a cell’s cycle life.
Takeaways & Limitations
State-informed hybrid models can provide accurate voltage prediction with relatively simple structures and potential utility across high-power and other LiB energy-storage applications.
Takeaways & Limitations
The physical-model state is a reduced-order representation that cannot exactly reconstruct the full state, requiring an assumption of an approximate nonlinear transformation.
Abstract
from arXiv · showhide
Mathematical modeling of lithium-ion batteries (LiBs) is a primary challenge in advanced battery management. This paper proposes two new frameworks to integrate physics-based models with machine learning to achieve high-precision modeling for LiBs. The frameworks are characterized by informing the machine learning model of the state information of the physical model, enabling a deep integration between physics and machine learning. Based on the frameworks, a series of hybrid models are constructed, through combining an electrochemical model and an equivalent circuit model, respectively, with a feedforward neural network. The hybrid models are relatively parsimonious in structure and can provide considerable voltage predictive accuracy under a broad range of C-rates, as shown by extensive simulations and experiments. The study further expands to conduct aging-aware hybrid modeling, leading to the design of a hybrid model conscious of the state-of-health to make prediction. The experiments show that the model has high voltage predictive accuracy throughout a LiB's cycle life.
1. Introduction
Accurate and efficient LiB models are needed for battery management, but physics-based and data-driven approaches each have important limitations. The paper addresses this gap with hybrid frameworks that feed physical-model state information into neural networks and validates models spanning broad C-rates and aging.
- Accurate and computationally efficient LiB models support state estimation, control, and diagnosis across expanding energy-storage applications.
- Electrochemical models can be accurate but computationally complex, while simplified models such as the SPM lose accuracy above low to medium C-rates.
- Equivalent circuit models offer parsimonious, computation-friendly structures, but their simplicity restricts accuracy and typically limits them to low to medium C-rates.
- Physics-based models and ML are complementary: physics provides interpretation and extrapolation, whereas ML captures uncertain nonlinear relationships and can run quickly after training.
- Existing hybrid models remained limited because neural networks learned mappings that were not physically meaningful and could predict poorly despite satisfactory training accuracy.
- HYBRID-1 captures physical-model residuals and HYBRID-2 predicts terminal voltage, with both frameworks feeding physical-model state information to an FNN.
- SPMT- and NDC-based hybrids achieved high voltage predictive accuracy across broad C-rate ranges, while an aging-aware model used SoH information to predict throughout cycle life.
- The proposed models are reported to generalize beyond training datasets and may support high-C-rate and high-power LiB applications.
2. Hybrid Physics-ML Modeling for LiBs
The paper proposes two hybrid physics-ML frameworks in which an FNN receives physical-model state information, enabling it to learn either physical-model voltage residuals or terminal voltage directly. The frameworks combine physical models with FNNs while supporting modular extensions such as electrochemical or equivalent-circuit components and aging-aware prediction.
- Proposed frameworks: HYBRID-1 cascades a physical model with an FNN that learns residual voltage error, while HYBRID-2 trains the FNN to predict terminal voltage directly.Both frameworks operate the physical model and FNN simultaneously, with the FNN informed by physical-model state information.
- Proposed frameworks: The FNN takes the physical model’s state X and applied current I as inputs, allowing it to learn mappings to residual voltage or true voltage.The proposed input choice is motivated by the physical model’s reduced-order representation of the battery’s full dynamics.
- Proposed frameworks: The physical model state is the pivotal difference from prior hybrid designs because it supports learning relationships intended to remain physically consistent.The paper argues that tighter physics-ML integration enhances predictive accuracy.
- Framework extensions: The frameworks are modular: either an electrochemical model or an equivalent circuit model can serve as the physical component.The paper demonstrates this flexibility with SPMT- and NDC-based hybrid models and extends the formulation toward state-of-health-aware prediction.
- FNN model: An FNN consists of serial input, hidden, and output layers, with nonlinear transformations in hidden layers and forward-only information flow.The paper presents FNNs as flexible function approximators trained by learning network parameters from measurement data, commonly through mean-squared-error optimization and stochastic gradient methods.
3. Hybrid Modeling via SPMT+FNN
SPMTNet-1 and SPMTNet-2 integrate the SPMT electrochemical-thermal model with an FNN informed by aggregated physical states. Simulations show accurate voltage prediction across training and testing conditions, outperforming the SPMT and existing hybrid models.
- Physical model: The SPMT model represents electrode particles and lithium-ion diffusion while incorporating thermal dynamics through temperature-dependent transport and heat-balance relations.Its computational speed comes with reduced accuracy at high C-rates or under uncertainty.
- Model construction: SPMTNet-1 and SPMTNet-2 combine the SPMT model with an FNN, using SoCbulk, SoCsurf, and temperature as physics-informed inputs.SPMTNet-1 learns the residual voltage, whereas SPMTNet-2 approximates terminal voltage directly.
- Model construction: Several aggregated physical states suffice for the FNN, reducing training and computational demands compared with using the full SPMT state.The study identifies SoCbulk, SoCsurf, and T as a favorable balance between efficiency and prediction performance.
- Simulation validation: SPMTNet-1 retains high testing accuracy and consistently outperforms the SPMT, particularly at medium to very high currents.The evaluation uses RMSE and relative error reduction across constant-current and driving-cycle datasets.
- Simulation validation: The proposed SPMT hybrid models provide higher testing accuracy and better voltage prediction than existing LiB hybrid models in extensive simulations.The reported result supports feeding physical-model state information into the ML model.
4. Hybrid Modeling via NDC+FNN
NDCNet-1 and NDCNet-2 combine an equivalent circuit model of diffusion and electrical processes with an FNN receiving the model’s internal states and temperature. Experiments show improved voltage prediction, especially at high currents, with parsimonious structures and better consistency than pure FNNs.
- Physical model: The NDC model represents electrode diffusion with coupled R-C pairs and simultaneously models nonlinear voltage dynamics.Its bulk and surface branches encode distinct electrode regions and lithium-ion charge transfer.
- Model construction: NDCNet-1 learns the NDC model’s residual voltage, while NDCNet-2 learns terminal voltage directly from NDC states and temperature.Both models feed Vb, Vs, V1, and T into the FNN.
- Experimental validation: Both NDC hybrid models substantially reduce RMSE relative to the NDC model, especially at high C-rates, while retaining high testing accuracy.Their voltage predictions improve consistently under LA92 and UDDS profiles, with larger gains at high currents.
- Model construction: NDCNet-1 and NDCNet-2 are more parsimonious than the SPMT-based hybrids, making them potentially more suitable for computation and real-world applications.This follows from the simpler structure of the NDC physical model.
- Experimental validation: NDCNet-1 versions with fewer neurons outperform four pure FNN models in training and testing and show better consistency across profiles.Pure FNN performance varies substantially between LA92 and UDDS, whereas the hybrid design remains more consistent.
5. Aging-Aware Hybrid Modeling
The aging-aware AA-NDCNet-1 extends the NDCNet-1 by supplying state-of-health as an FNN input, enabling voltage prediction across degradation and cells.
- Motivation: LiB aging changes material properties, capacity, resistance, and heat behavior, making the battery a time-varying system that is difficult to model.The difficulty includes characterizing physical relationships between aging and battery behavior.
- Model design: SoH is defined as the ratio of current capacity Qa to initial capacity Qinit and is used as a concise aging indicator.The paper uses this definition to preserve model parsimony.
- Model design: AA-NDCNet-1 feeds regularly calculated SoH into the FNN, while leaving the NDC model unchanged to capture aging effects through the FNN.This design avoids continually updating the NDC model based on aging condition.
- Experimental validation: The evaluation used two Samsung INR18650-25R cells cycled 450 times until capacity reached about 81% of its initial value.Cell #1 supplied training data and cell #2 supplied testing data across constant and variable current profiles.
- Results: AA-NDCNet-1 achieved RMSE almost consistently below 20 mV across tested SoH values, while NDC-model RMSE increased steadily with aging.The model retained high accuracy across the aging process and predicted cell #2 throughout its cycle life despite training on cell #1.
6. Discussion
The discussion finds that physics-informed hybrid models combine accurate, physically consistent voltage prediction with computational efficiency and broad operating-range applicability.
- Predictive accuracy: Feeding physical-model state information into the FNN supports accurate and physically consistent prediction, provided training data cover intended C-rates, SoC, and SoH ranges.The discussion identifies both physics-informed inputs and representative data coverage as key factors.
- Computational efficiency: Hybrid-model training cost depends on data quantity and FNN structure, but simpler FNNs than pure data-driven models reduce training cost and enable fast online prediction.The reported experience places NDC-based hybrids above SPMT-based hybrids in computational efficiency, with both faster than the DFN in the cited comparison.
- Prospective applications: Wide C-rate voltage prediction makes the hybrid models suitable for practical LiB storage systems, including high-power applications where conventional models face cost or accuracy constraints.The discussion specifically identifies electric aircraft as an example operating up to 5 C during take-off and landing.
- Prospective applications: For low- to medium-C-rate systems, an FNN can complement a high-fidelity physical model by capturing uncertain nonlinear voltage for voltage prediction and SoC estimation.The proposed approach is therefore described as relevant beyond high-power systems.
- FNN architecture selection: Practitioners can choose smaller FNN architectures when comparable performance permits a trade-off between prediction accuracy and model size or complexity.The validation used a four-layer FNN with two 32-neuron hidden layers, but fewer neurons could perform comparably.
7. Conclusion
The paper develops physics-informed hybrid frameworks that combine physical models with machine learning for accurate, efficient LiB voltage prediction across operating conditions and aging.
- Conclusion: HYBRID-1 and HYBRID-2 inform machine learning models of physical-model state information to improve voltage prediction and simplify ML architecture.The frameworks integrate electrochemical models and equivalent circuit models with feedforward neural networks.
- Conclusion: Four SPMT+FNN and NDC+FNN hybrid models achieved exceptionally high voltage predictive accuracy across a wide range of C-rates in simulations and experiments.The conclusion reports this as the central validation outcome of the proposed frameworks.
- Conclusion: An NDC+FNN model was extended with state-of-health awareness and experimentally validated for accurate prediction across a LiB’s cycle life.The extension makes the ML component informed of the battery’s aging condition.