# Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach

arXiv: https://arxiv.org/abs/2506.05245 · HTML: https://mlsanity.com/paper/2506.05245/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 57 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Learning Continuous System Dynamics from Irregularly-Sampled Partial Observations](https://mlsanity.com/paper/2011.03880/) (2020)
- [Machine-learning hidden symmetries](https://mlsanity.com/paper/2109.09721/) (2021)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Neural Ordinary Differential Equation Control of Dynamics on Graphs](https://mlsanity.com/paper/2006.09773/) (2020)
- [AI Poincar'e 2.0: Machine Learning Conservation Laws from Differential Equations](https://mlsanity.com/paper/2203.12610/) (2022)
- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [Meta-learning Pseudo-differential Operators with Deep Neural Networks](https://mlsanity.com/paper/1906.06782/) (2019)
- [A Penalty-free Infeasible Approach for a Class of Nonsmooth Optimization Problems over the Stiefel Manifold](https://mlsanity.com/paper/2103.03514/) (2021)
- [Explainable Deep Learning for Tumor Dynamic Modeling and Overall Survival Prediction using Neural-ODE](https://mlsanity.com/paper/2308.01362/) (2023)
- [Learning quantum dynamics with latent neural ODEs](https://mlsanity.com/paper/2110.10721/) (2021)
- [Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators](https://mlsanity.com/paper/2503.00645/) (2025)

## Built on by (most cited first, 1 of 1)

- [Learning Effective Soliton Dynamics from Scattering Data](https://mlsanity.com/paper/2607.01545/) (2026)

## Abstract (arXiv)

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"odinger 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.