Sparse-to-Field Reconstruction via Stochastic Neural Dynamic Mode Decomposition

TL;DR

This paper introduces Stochastic NODE-DMD, a probabilistic method that models nonlinear continuous-time dynamics from sparse data, enabling accurate reconstruction and uncertainty quantification in complex physical systems.

Contribution

It presents a novel probabilistic extension of DMD that captures nonlinear dynamics, provides uncertainty estimates, and improves reconstruction from limited observations.

Findings

Outperforms baseline in reconstruction accuracy with only 10% observations.

Recovers dynamical structure by aligning modes and eigenvalues with ground truth.

Learns calibrated distributions over dynamics, preserving ensemble variability.

Abstract

Many consequential real-world systems, like wind fields and ocean currents, are dynamic and hard to model. Learning their governing dynamics remains a central challenge in scientific machine learning. Dynamic Mode Decomposition (DMD) provides a simple, data-driven approximation, but practical use is limited by sparse/noisy observations from continuous fields, reliance on linear approximations, and the lack of principled uncertainty quantification. To address these issues, we introduce Stochastic NODE-DMD, a probabilistic extension of DMD that models continuous-time, nonlinear dynamics while remaining interpretable. Our approach enables continuous spatiotemporal reconstruction at arbitrary coordinates and quantifies predictive uncertainty. Across four benchmarks, a synthetic setting and three physics-based flows, it surpasses a baseline in reconstruction accuracy when trained from only 10% observation density. It further recovers the dynamical structure by aligning learned modes and continuous-time eigenvalues with ground truth. Finally, on datasets with multiple realizations, our method learns a calibrated distribution over latent dynamics that preserves ensemble variability rather than averaging across regimes. Our code is available at: https://github.com/sedan-group/Stochastic-NODE-DMD