Physics-Informed Neural Network with Transfer Learning for State Estimation in Lithium-Ion Batteries using the Single Particle Model with Electrolyte

Source: arXiv:2606.28220 · Published 2026-06-26 · By Gift Modekwe, Qiugang Lu

TL;DR

This paper addresses the computational inefficiency and limited generalization of physics-informed neural networks (PINNs) applied to the electrochemical modeling of lithium-ion batteries, specifically using the single particle model with electrolyte (SPMe). While PINNs enforce the governing PDEs of battery physics within the loss function to produce physically consistent state estimates, training them from scratch for each new battery type or operating condition is slow and challenging. The authors propose integrating transfer learning into an SPMe-based PINN framework to leverage pretrained electrochemical knowledge from a source battery, then adapt the model efficiently to different target batteries via fine-tuning and selective freezing of network layers. This approach significantly reduces training time, improves convergence stability, and enables accurate voltage and internal state prediction across various battery chemistries and formats. Validation on publicly available datasets and comparison against PyBaMM reference solutions demonstrate low prediction errors (e.g., RMSE 8.1e-4 on the source battery) and strong parameter estimation fidelity for solid-phase diffusivities. The study establishes transfer learning as a promising strategy for scalable, real-time physics-informed battery state estimation.

Key findings

• PINN trained from scratch on source battery B1 (LG M50 cell) predicts terminal voltage with RMSE of 8.1e-4, closely matching reference PyBaMM results.
• Transfer learning adaptation to domain-shifted battery B2 (Kokam pouch cell, graphite/NMC chemistry) yields accurate terminal voltage and internal surface lithium concentrations with minimal error increase.
• Robust terminal voltage prediction across discharge rates: 1C and 0.5C tested on B2 after transfer learning.
• Successful cross-chemistry transfer to B3 (LFP cell) with accurate voltage prediction at 1C and 1.2C discharges, despite distinct electrochemical characteristics.
• Learned solid-phase diffusivities for both electrodes closely match true values in B2 and B3 datasets, validating parameter identification capability.
• Freezing the first two layers during fine-tuning preserves general electrochemical representations while allowing efficient adaptation to target battery specifics.
• Transfer learning significantly reduces PINN training time and improves convergence stability compared to training from scratch (exact speed-up numbers not specified).
• Voltage prediction incorporates fully differentiable SPMe voltage equations enabling joint optimization of neural network parameters and electrochemical kinetic parameters.

Threat model

n/a — this paper is focused on physics-informed machine learning for battery modeling rather than adversarial or security threat scenarios. The implicit assumption is a well-behaved environment where governing physical laws hold and training data is trustworthy.

Methodology — deep read

1. Threat Model & Assumptions: The adversary context is not security focused; the framework assumes access to source battery data and some target battery measurements (voltage and states) to enable transfer learning. The assumption is that physics (SPMe PDEs) hold across batteries but parameter variations require adaptation.

2. Data: The pretrained model is trained on synthetic electrochemical data generated from the SPMe with parameter set B1 (LG M50 cell; cylindrical graphite/NMC). Target battery data sets include B2 (Kokam pouch cell graphite/NMC) and B3 (LFP cell) with labeled voltage measurements. Operating conditions include constant current discharges at rates from 0.5C to 1.2C. Data preprocessing normalizes inputs (time, spatial coordinates) and concentrations to max values.

3. Architecture/Algorithm: The PINN framework contains three neural networks approximating negative electrode solid-phase, positive electrode solid-phase, and electrolyte concentration states. Each network uses four hidden layers with 64 neurons per layer, taking normalized spatio-temporal coordinates as input and predicting concentration states. The composite loss includes PDE residuals enforcing SPMe governing equations, boundary conditions, initial conditions, and voltage mismatch to reference PyBaMM data. Electrochemical parameters such as solid diffusivities are treated as learnable parameters.

4. Training Regime: In pretraining, the model optimizes all network parameters and kinetic parameters to minimize the weighted composite loss across collocation points and observed voltage data using backpropagation. For fine-tuning on target batteries, pretrained parameters initialize the network; selective freezing of early layers preserves learned physics-based representations, while later layers and electrochemical parameters are optimized based on target battery data. The weighting factor balances PDE residual and data fidelity losses. Hyperparameters such as batch size, epochs, optimizer type, and learning rate are not explicitly detailed.

5. Evaluation Protocol: Model predictions of terminal voltage and internal electrochemical states (surface solid-phase lithium concentrations) are compared to PyBaMM SPMe model outputs as ground truth. The primary metric for voltage prediction is root mean square error (RMSE). Parameter estimation quality is assessed by comparing learned diffusivities to true values from datasets. Tests include multiple discharge rates and both cross-domain (similar chemistry but different geometry) and cross-chemistry transfers. Ablations involve freezing first two layers during fine-tuning.

6. Reproducibility: There is no explicit mention of public code release or frozen weights. The datasets used (B1, B2, B3) are based on publicly documented parameter sets and simulation data from prior published literature, such as Chen et al. (2020), Ecker et al. (2015), and Prada et al. (2013). The training pipeline is described sufficiently for replication given access to these datasets and the PyBaMM simulation environment.

Example End-to-End: For the B2 battery, the pretrained SPMe-PINN weights from B1 are loaded. The first two layers are frozen, and the remaining layers along with solid-phase diffusivities are optimized using 1C discharge voltage measurements alongside PDE residual losses to fine-tune the model. The final adapted model accurately predicts terminal voltage traces and internal lithium concentrations close to the reference PyBaMM simulations, demonstrating effective transfer learning in physics-informed battery modeling.

Technical innovations

• Integration of transfer learning into a physics-informed neural network framework based on the SPMe model for lithium-ion battery state estimation.
• Joint optimization of neural network weights and key electrochemical parameters (e.g., solid-phase diffusivities) within a PINN framework to enable parameter estimation during fine-tuning.
• Selective freezing strategy during fine-tuning that preserves shared electrochemical knowledge while adapting to new battery chemistries or geometries.
• Use of fully differentiable SPMe voltage equations embedded in the PINN loss function to backpropagate voltage errors through predicted concentration fields.

Datasets

• B1 (LG M50 graphite/NMC cylindrical cell) — size not explicitly stated — synthetic data generated from PyBaMM simulation
• B2 (Kokam graphite/NMC pouch cell) — size not explicitly stated — based on Ecker et al. (2015) parameter set
• B3 (LFP cell) — size not explicitly stated — based on Prada et al. (2013) parameter set

Baselines vs proposed

• Pretrained PINN B1: terminal voltage RMSE = 8.1e-4; compared to PyBaMM reference
• Transfer learning on B2 at 1C discharge: voltage predictions closely track PyBaMM with minor error increases vs B1 baseline
• Transfer learning on B2 at 0.5C discharge: maintained high voltage prediction accuracy vs PyBaMM
• Transfer learning cross-chemistry to B3 at 1C and 1.2C: robust voltage predictions matching PyBaMM across discharge rates
• Solid-phase diffusivity estimation: Predicted vs True diffusivities differ by <0.1% across B2 and B3 batteries (e.g., B2 negative electrode: 8.341e-15 predicted vs 8.332e-15 true)

Limitations

• The approach relies on availability of reference voltage or simulation data for target batteries during fine-tuning; limited discussion on fully unsupervised adaptation.
• Training hyperparameters and optimizer details are not fully disclosed, impeding exact reproducibility and performance benchmarking.
• No explicit runtime or computational cost numbers provided to quantify training speed-up from transfer learning versus training from scratch.
• Validation is limited to simulated datasets and a few standardized battery parameter sets; no evaluation on real experimental datasets with noise or sensor errors.
• Thermal effects and battery aging/degradation dynamics are not modeled; future work planned to incorporate these important practical factors.
• No adversarial robustness or sensitivity analysis to distribution shifts beyond the tested chemistries/operating conditions.

Open questions / follow-ons

• How robust is the transfer learning framework when target battery data is noisy, sparse, or limited compared to simulation?
• Can the PINN model effectively incorporate battery aging processes and temperature dependence for long-term state estimation?
• What are the computational speed-ups and energy efficiency gains in real embedded battery management systems?
• Is it possible to extend this framework for online continual learning to handle dynamically changing battery conditions in the field?

Why it matters for bot defense

From a bot-defense and CAPTCHA perspective, this work exemplifies how physics-informed transfer learning can efficiently generalize complex PDE-governed systems across different domains using limited new data. Practitioners designing CAPTCHA or bot-detection models based on physics-informed or hybrid machine learning may consider transfer learning strategies like layer freezing and parameter fine-tuning to adapt models across environments or threat scenarios without retraining from scratch. The PINN framework here leverages differentiable physics constraints embedded in loss functions, illustrating how strong priors can stabilize learning and reduce data requirements while preserving interpretability—concepts potentially transferable to modeling human-bot interaction dynamics with underlying system constraints. However, the paper’s domain is distinct, focusing on electrochemical battery modeling rather than security, so direct application to CAPTCHA defenses would require adaptation of the underlying physics and threat models.

Cite

@article{arxiv2606_28220,
  title={ Physics-Informed Neural Network with Transfer Learning for State Estimation in Lithium-Ion Batteries using the Single Particle Model with Electrolyte },
  author={ Gift Modekwe and Qiugang Lu },
  journal={arXiv preprint arXiv:2606.28220},
  year={ 2026 },
  url={https://arxiv.org/abs/2606.28220}
}

Read the full paper

• arXiv:2606.28220 (abstract)
• arXiv:2606.28220 (PDF)