Factorized Neural Implicit DMD for Parametric Dynamics

2026-03-11

AI Summary

This work addresses the challenge of modeling high-dimensional nonlinear physical systems in the absence of explicit governing equations by proposing a novel approach that integrates neural implicit fields with spectral decomposition of the Koopman operator. By factorizing and decoupling spatial modes from temporal dynamics, the method constructs a generalizable, parameterized flow operator capable of explicitly learning the system’s spectral structure without requiring prior knowledge of the underlying dynamics. It represents the first integration of neural implicit representations with dynamic mode decomposition, enabling stable long-term prediction, interpolation across parameters, and accurate identification of eigenmodes, eigenvalues, and stability characteristics. The framework demonstrates high accuracy and strong generalization across diverse spatiotemporal dynamical systems.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

A data-driven, model-free approach to modeling the temporal evolution of physical systems mitigates the need for explicit knowledge of the governing equations. Even when physical priors such as partial differential equations are available, such systems often reside in high-dimensional state spaces and exhibit nonlinear dynamics, making traditional numerical solvers computationally expensive and ill-suited for real-time analysis and control. Consider the problem of learning a parametric flow of a dynamical system: with an initial field and a set of physical parameters, we aim to predict the system's evolution over time in a way that supports long-horizon rollouts, generalization to unseen parameters, and spectral analysis. We propose a physics-coded neural field parameterization of the Koopman operator's spectral decomposition. Unlike a physics-constrained neural field, which fits a single solution surface, and neural operators, which directly approximate the solution operator at fixed time horizons, our model learns a factorized flow operator that decouples spatial modes and temporal evolution. This structure exposes underlying eigenvalues, modes, and stability of the underlying physical process to enable stable long-term rollouts, interpolation across parameter spaces, and spectral analysis. We demonstrate the efficacy of our method on a range of dynamics problems, showcasing its ability to accurately predict complex spatiotemporal phenomena while providing insights into the system's dynamic behavior.

Problem

• parametric dynamics
• Koopman operator
• neural implicit representation
• spatiotemporal modeling
• spectral analysis

Innovation

• Factorized Neural Implicit
• Koopman Operator
• Spectral Decomposition
• Parametric Dynamics
• Neural Field