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IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments. MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically. Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.
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This content will become publicly available on December 1, 2026
Robust moment identification for nonlinear PDEs via a neural ODE approach
We propose a data-driven framework for learning reduced-order moment dynamics from partial differential equation (PDE)-governed systems using neural Ordinary Differential Equations (neural ODEs). In contrast to derivative-based methods, such as SINDy (Sparse Identification of Nonlinear Dynamics), 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 Schrödinger equation, the framework accurately recovers governing moment dynamics when closure is available, even from limited, irregular, and noisy 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–Kolmogorov–Petrovskii–Piskounov 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.
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- Award ID(s):
- 2408988
- PAR ID:
- 10690650
- Publisher / Repository:
- AIP
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 35
- Issue:
- 12
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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