Multi-level Constraints in PINNS: Theory and Applications to Battery Impedance Modeling

Accurate estimation of battery state-of-health (SOH) from electrochemical impedance spectroscopy (EIS) is crucial for safe and long-lived battery operation. Purely data-driven models, however, often struggle under limited, noisy, or out-of-distribution data. Hybrid physics-data models, including physics-informed neural networks (PINNs), can enhance robustness by incorporating physical priors; yet, systematic guidance on which priors are most beneficial, when, and why remains limited. We introduce a practical taxonomy that classifies priors into system-level constraints (S), which encode parametric, mechanism-level structure (e.g., equivalent-circuit models, Butler-Volmer relations, PDEs), and general-consistency constraints (G), which enforce model-agnostic validity (e.g., Kramers-Kronig relations, monotonicity). Using a constrained-hypothesis-space framework, we analyze how S and G affect identifiability and generalization, and we derive, under mild assumptions and compatibility conditions, an expected ordering of performance across constraint combinations. We also identify characteristic failure modes arising from overconstrained or incompatible priors. Empirically, we validate our theoretical insights using a public EIS dataset, evaluating SOH prediction with RMSE and MAE. Our experiments also evaluate several constraint combinations that have not been explored in prior PINN-based battery studies, enabling a systematic comparison across multi-prior configurations.