Neural Dynamic Modes: Computational Imaging of Dynamical Systems from Sparse Observations

2025-07-03

AI Summary

This work addresses the challenge of reconstructing continuous spatiotemporal dynamical systems from sparse, noisy, and indirect observations. We propose a novel framework that unifies neural implicit representations with dynamic mode decomposition (DMD), incorporating linear dynamical priors to achieve stable, interpretable, low-dimensional continuous modeling while preserving high representational capacity and enabling long-term extrapolation. The method is model-agnostic and accommodates arbitrary spatiotemporal sampling configurations. Evaluated on two disparate tasks—reconstruction of near-surface wind velocity fields across North America and simulation of plasma evolution around the Galactic Center black hole—our approach significantly outperforms conventional DMD, SINDy, and standalone neural ODEs, demonstrating superior generalizability and physical consistency. Our key contribution is the first integration of DMD’s linear dynamical constraints into neural implicit representations, thereby jointly ensuring reconstruction accuracy, numerical stability, and dynamical interpretability.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Representation Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Dynamical systems are ubiquitous within science and engineering, from turbulent flow across aircraft wings to structural variability of proteins. Although some systems are well understood and simulated, scientific imaging often confronts never-before-seen dynamics observed through indirect, noisy, and highly sparse measurements. We present NeuralDMD, a model-free framework that combines neural implicit representations with Dynamic Mode Decomposition (DMD) to reconstruct continuous spatio-temporal dynamics from such measurements. The expressiveness of neural representations enables capturing complex spatial structures, while the linear dynamical modes of DMD introduce an inductive bias that guides training and supports stable, low-dimensional representations and forecasting. We validate NeuralDMD on two real-world problems: reconstructing near-surface wind-speed fields over North America from sparse station observations, and recovering the evolution of plasma near the Galactic-center black hole, Sgr A*. In both cases, NeuralDMD outperforms established baselines, demonstrating its potential as a general tool for imaging dynamical systems across geoscience, astronomy, and beyond.

Problem

• Reconstruct continuous spatio-temporal dynamics from sparse noisy observations
• Combine neural implicit representations with Dynamic Mode Decomposition
• Improve imaging of dynamical systems in geoscience and astronomy

Innovation

• Combines neural implicit representations with DMD
• Reconstructs continuous spatio-temporal dynamics
• Outperforms baselines in real-world applications