Impact of Physics-Informed Features on Neural Network Complexity for Li-ion Battery Voltage Prediction in Electric Vertical Takeoff and Landing Aircrafts

The electrification of vertical takeoff and landing aircraft demands high-fidelity battery management systems capable of predicting voltage response under aggressive power dynamics. While data-driven models offer high accuracy, they often require complex architectures and extensive training data. Conversely, equivalent circuit models (ECMs), such as the second-order model, offer physical interpretability but struggle with high C-rate non-linearities. This paper investigates the impact of integrating physics-based information into data-driven surrogate models. Specifically, we evaluate whether physics-informed features allow for the simplification of neural network architectures without compromising accuracy. Using the open-source electric vertical takeoff and landing (eVTOL) battery dataset, we compare pure data-driven models against physics-informed data models. Results demonstrate that physics-informed models achieve comparable accuracy to complex pure data-driven models while using up to 75% fewer trainable parameters, significantly reducing computational overhead for potential on-board deployment.

💡 Research Summary

The paper addresses the pressing need for accurate, real‑time voltage prediction of lithium‑ion batteries powering electric vertical take‑off and landing (eVTOL) aircraft. eVTOL missions are characterized by short, high‑power pulses during take‑off, cruise, and landing, which cause rapid voltage drops and significant thermal stress. Traditional physics‑based models such as the pseudo‑two‑dimensional (P2D) model provide high fidelity but are computationally prohibitive for onboard battery management systems (BMS). Equivalent circuit models (ECMs), especially the second‑order 2RC model, are computationally cheap and interpretable but struggle to capture the strong non‑linearities that arise at high C‑rates typical of eVTOL operation. Data‑driven deep neural networks (DNNs) have demonstrated strong performance in capturing complex battery dynamics, yet they usually require deep architectures, large training datasets, and substantial inference resources, making them unsuitable for edge‑AI constrained BMS hardware. Moreover, pure data‑driven models often exhibit poor extrapolation when operating outside the training domain.

To bridge the gap between physics‑based interpretability and data‑driven flexibility, the authors propose a physics‑informed machine learning (PIML) framework that embeds the 2RC ECM as a prior within a neural network. The approach, termed a Physics‑Informed Neural Network (PINN), computes a physics‑based voltage estimate V_phy at each time step using the discrete‑time state‑space equations of the 2RC model. The neural network is then tasked only with learning the residual correction ΔV_θ, i.e., the difference between the measured terminal voltage and V_phy. By anchoring the prediction to a physically meaningful baseline, the hypothesis space of the network is dramatically reduced, allowing for much simpler architectures without sacrificing accuracy.

The authors evaluate the method using the open‑source eVTOL battery dataset released by Bills et al. (2023). Five cells are considered; four (VAH05, VAH10, VAH12, VAH26) form the training set, while the fifth (VAH11) serves as an unseen test case. Training data are sampled from cycles 1, 50, and 1000 to expose the models to a wide range of aging states and mission profiles (different power reductions, temperature conditions, and cruise times). Input features for the pure data‑driven Feed‑Forward Neural Network (FNN) consist of Δt, current I, state‑of‑charge (SOC), temperature T, and cycle number N. For the PINN, the feature vector is augmented with physics‑derived quantities: open‑circuit voltage (OCV), the two RC polarization states (V_RC1, V_RC2), and the physics‑based voltage V_phy itself.

A systematic architecture sweep is performed: hidden layers = {1, 2, 4} and neurons per layer = {32, 64, 128}. Both models are trained to minimize mean‑squared error (MSE) between predicted and measured voltages. Performance metrics include maximum error, mean absolute error (MAE), root‑mean‑square error (RMSE), R², and inference time.

Key findings:

1. Model Compression – The simplest PINN (1 hidden layer, 32 neurons) achieves RMSE = 27.7 mV, MAE = 15.9 mV, and R² = 98.5 %, while the equivalent‑size FNN records RMSE = 61.0 mV, MAE = 51.4 mV, and R² = 92.8 %. Thus, the physics‑informed approach halves the error with the same network size.

2. Optimal Trade‑off – The best performing PINN (2 hidden layers, 64 neurons) reaches RMSE = 20.1 mV, MAE = 9.65 mV, and R² = 99.2 % using only ~192 trainable parameters. A deeper FNN (4 layers, 128 neurons) still lags behind with RMSE = 56.7 mV, MAE = 39.2 mV, and R² = 93.8 %.

3. Peak‑Error Reduction – Maximum voltage error for PINNs stays below 110 mV, whereas the FNN’s peak error can exceed 300 mV, indicating superior robustness during rapid transients.

4. Parameter Savings – Across the sweep, PINNs consistently require up to 75 % fewer trainable parameters than their FNN counterparts to achieve comparable or better accuracy, directly translating into lower memory footprint and faster inference—critical for embedded BMS hardware.

The authors attribute these gains to the residual learning formulation: the 2RC model already captures the dominant linear dynamics, leaving the neural network to model only the non‑linear residuals. Consequently, the network’s learning task is simpler, reducing over‑fitting risk and data hunger. Moreover, the inclusion of physics‑derived features (RC states, OCV) provides the network with explicit information about the battery’s internal state, further easing the learning burden.

Limitations and future work are acknowledged. The 2RC model, while computationally cheap, does not fully represent high‑C‑rate non‑linearities, meaning the residual term can still be sizable. Future research could replace the 2RC prior with more sophisticated physics models (e.g., P2D, Single‑Particle Model) or adopt multi‑task learning to jointly estimate SOC, SOH, and voltage. Extending validation to other chemistries, wider temperature ranges, and real‑time flight tests would solidify the approach’s generality.

In conclusion, the paper demonstrates that embedding a simple physics‑based equivalent circuit model into a neural network dramatically reduces model complexity while preserving, and even enhancing, prediction fidelity for eVTOL battery voltage. This physics‑informed residual learning paradigm offers a practical pathway toward lightweight, high‑accuracy surrogate models suitable for the stringent computational constraints of onboard battery management and flight control systems.