Physics-Informed Neural ODEs with Scale-Aware Residuals for Learning Stiff Biophysical Dynamics

2025/11/13 by Kamalpreet Singh Kainth, Kainth, Kamalpreet Singh, Paritosh Joshi +7
Computer Science · Mathematics · Physics and Astronomy · #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Reservoir Computing #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2511.11734

openalex publication_date 2025/11/13 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

Neural differential equations offer a powerful framework for modeling continuous-time dynamics, but forecasting stiff biophysical systems remains unreliable. Standard Neural ODEs and physics informed variants often require orders of magnitude more iterations, and even then may converge to suboptimal solutions that fail to preserve oscillatory frequency or amplitude. We introduce PhysicsInformed Neural ODEs with with Scale-Aware Residuals (PI-NODE-SR), a framework that combines a low-order explicit solver (Heun method) residual normalisation to balance contributions between state variables evolving on disparate timescales. This combination stabilises training under realistic iteration budgets and avoids reliance on computationally expensive implicit solvers. On the Hodgkin-Huxley equations, PI-NODE-SR learns from a single oscillation simulated with a stiff solver (Rodas5P) and extrapolates beyond 100 ms, capturing both oscillation frequency and near-correct amplitudes. Remarkably, end-to-end learning of the vector field enables PI-NODE-SR to recover morphological features such as sharp subthreshold curvature in gating variables that are typically reserved for higher-order solvers, suggesting that neural correction can offset numerical diffusion. While performance remains sensitive to initialisation, PI-NODE-SR consistently reduces long-horizon errors relative to baseline Neural-ODEs and PINNs, offering a principled route to stable and efficient learning of stiff biological dynamics.

Citations

• Universal differential equations for systems biology: Current state and open problems
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