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

arXiv: https://arxiv.org/abs/2511.11734 · HTML: https://mlsanity.com/paper/2511.11734/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 23 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Builds on (most cited first, 12 of 12)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Proximal Implicit ODE Solvers for Accelerating Learning Neural ODEs](https://mlsanity.com/paper/2204.08621/) (2022)
- [Generative time series models using Neural ODE in Variational Autoencoders](https://mlsanity.com/paper/2201.04630/) (2022)
- [Estimate Epidemiological Parameters given Partial Observations based on Algebraically Observable PINNs](https://mlsanity.com/paper/2407.12598/) (2024)

## Built on by (most cited first, 2 of 2)

- [Learning Hybrid Biophysical Neuron Models with Neural ODEs](https://mlsanity.com/paper/2606.16693/) (2026)
- [Robust Parameter and State Estimation in Multiscale Neuronal Systems Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2603.08742/) (2026)

## Abstract (arXiv)

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.