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

TL;DR

Stochastic NODE-DMD addresses the challenge of reconstructing and forecasting high-dimensional, dynamic fields from sparse, noisy observations by fusing a probabilistic DMD framework with a continuous-time neural ODE and a neural implicit spatial encoder. It yields uncertainty quantification for both measurements and latent dynamics and supports grid-free prediction at arbitrary coordinates. The method preserves spectral interpretability while extending applicability to nonlinear, sparse data, and it learns calibrated distributions across multiple realizations rather than collapsing to a single regime. Evaluations on synthetic and physics-based flows demonstrate robust reconstruction, improved long-horizon stability, and grid-density independence, with available code.

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

Figures (5)

• Figure 1: Overview of the Stochastic NODE--DMD architecture. At time step $k$, subsampled measurements $\mathbf{y}_k$ and set of spatial coordinates $S$ are fed into the Latent Encoder, which outputs eigenvalues $\Lambda$ and the latent distribution $p(\boldsymbol{\phi}_k)$ . The Neural ODE evolves this distribution forward to time $k+1$. The predicted latent state distribution $p(\boldsymbol{\hat{\phi}}_{k+1})$ is then combined with spatial modes from the Mode Extractor$W$ to reconstruct the distribution of subsampled field at time $k+1$, enabling uncertainty-aware, grid-free forecasting.
• Figure 2: Qualitative reconstruction results from sparse observations. Row labels list the task and rollout length $T$ (final time step). Columns: (1) training data—10% of fixed spatial points sampled from the ground truth, (2) ground-truth full field, (3) NODE--DMD 1-step (teacher-forced), (4) NODE--DMD $m$-step (autoregressive), and (5) NDMD $m$-step.
• Figure 3: Mode-portrait overlays at absolute levels. Colored contours show per-mode isocontours of $\lvert W_k(x)\,\phi_k(t)\rvert$ at thresholds fixed once per mode from the GT sequence (30/60/90-th percentiles). Backgrounds show the corresponding field (Left: ground-truth mode portrait; Right: model-predicted mode portrait).
• Figure 4: Particle trajectories from velocity field. From posterior samples (Red) vs. ground truth (Gray). Left: Model trained on a single vorticity-field realization. Right: Model trained on $10$ realizations.
• Figure 5: Reconstruction results at the final time step for low-resolution (50$\times$50) and high-resolution (200$\times$200) grids for the Gray--Scott system (first row), Vorticity flow (second row), and Cylinder flow (third row). Visualizations are generated using NDMD and the 1-step NODE--DMD model.