Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach

2025-06-05

AI Summary

Constructing robust low-dimensional moment dynamics models for nonlinear partial differential equation (PDE) systems remains challenging when only sparse, irregular, and noisy time-series data are available. Method: This paper proposes an end-to-end learning framework that avoids derivative estimation and prior closure assumptions. It introduces a novel Neural ODE-based paradigm for direct modeling of moment trajectories and incorporates a differentiable coordinate transformation constrained on the Stiefel manifold to automatically learn a closure-compatible low-dimensional moment representation. Results: Experiments on the nonlinear Schrödinger equation and the Fisher–KPP system demonstrate that the method significantly outperforms SINDy and expert-designed ODE models. It achieves high-accuracy moment dynamics identification and long-term predictive capability even under extreme sparsity (<10 time points) and high noise (SNR ≈ 5 dB). This work establishes a new data-driven paradigm for PDE model order reduction.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Learning reduced-order moment dynamics from PDE systems
• Robust modeling with sparse, irregular, noisy data
• Discovering low-dimensional representations for unclosed dynamics

Innovation

• Neural ODEs model moment trajectories robustly
• Stiefel manifold enables data-driven closure discovery
• Superior extrapolation accuracy with sparse data