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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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Data-driven model discovery and model selection for noisy biological systems
Biological systems exhibit complex dynamics that differential equations can often adeptly represent. Ordinary differential equation models are widespread; until recently their construction has required extensive prior knowledge of the system. Machine learning methods offer alternative means of model construction: differential equation models can be learnt from data via model discovery using sparse identification of nonlinear dynamics (SINDy). However, SINDy struggles with realistic levels of biological noise and is limited in its ability to incorporate prior knowledge of the system. We propose a data-driven framework for model discovery and model selection using hybrid dynamical systems: partial models containing missing terms. Neural networks are used to approximate the unknown dynamics of a system, enabling the denoising of the data while simultaneously learning the latent dynamics. Simulations from the fitted neural network are then used to infer models using sparse regression. We show, via model selection, that model discovery using hybrid dynamical systems outperforms alternative approaches. We find it possible to infer models correctly up to high levels of biological noise of different types. We demonstrate the potential to learn models from sparse, noisy data in application to a canonical cell state transition using data derived from single-cell transcriptomics. Overall, this approach provides a practical framework for model discovery in biology in cases where data are noisy and sparse, of particular utility when the underlying biological mechanisms are partially but incompletely known.
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- Award ID(s):
- 2045327
- PAR ID:
- 10573324
- Editor(s):
- Alber, Mark
- Publisher / Repository:
- PLOS
- Date Published:
- Journal Name:
- PLOS Computational Biology
- Volume:
- 21
- Issue:
- 1
- ISSN:
- 1553-7358
- Page Range / eLocation ID:
- e1012762
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1371/journal.pcbi.1012762
