Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach

TL;DR

This paper introduces a Neural ODE-based framework for robustly learning reduced-order moment dynamics from PDE systems, effective even with sparse, irregular, and unclosed data, demonstrated on nonlinear Schrödinger and reaction-diffusion equations.

Contribution

It presents a novel Neural ODE approach for moment dynamics, including a coordinate transformation for closure discovery and effective modeling without known closures.

Findings

Accurately recovers moment dynamics with limited data

Enables discovery of low-dimensional closed representations

Outperforms traditional models in unclosed, sparse data scenarios

Abstract

We propose a data-driven framework for learning reduced-order moment dynamics from PDE-governed systems using Neural ODEs. In contrast to derivative-based methods like SINDy, which necessitate densely sampled data and are sensitive to noise, our approach based on Neural ODEs directly models moment trajectories, enabling robust learning from sparse and potentially irregular time series. Using as an application platform the nonlinear Schr\"{o}dinger equation, the framework accurately recovers governing moment dynamics when closure is available, even with limited and irregular observations. For systems without analytical closure, we introduce a data-driven coordinate transformation strategy based on Stiefel manifold optimization, enabling the discovery of low-dimensional representations in which the moment dynamics become closed, facilitating interpretable and reliable modeling. We also explore cases where a closure model is not known, such as a Fisher-KPP reaction-diffusion system. Here we demonstrate that Neural ODEs can still effectively approximate the unclosed moment dynamics and achieve superior extrapolation accuracy compared to physical-expert-derived ODE models. This advantage remains robust even under sparse and irregular sampling, highlighting the method's robustness in data-limited settings. Our results highlight the Neural ODE framework as a powerful and flexible tool for learning interpretable, low-dimensional moment dynamics in complex PDE-governed systems.