KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra

2026-02-15

AI Summary

High-dimensional spatiotemporal chaotic systems are often dominated by continuous spectra, yet existing data-driven approaches frequently suffer from instability, limited interpretability, and poor scalability. This work proposes KoopGen—a generator-based neural Koopman framework that explicitly decomposes dynamics into conservative (skew-adjoint) and dissipative (self-adjoint) components via a state-dependent Koopman generator, while rigorously embedding operator-theoretic constraints. Notably, KoopGen achieves the first explicit separation of self-adjoint and skew-adjoint parts within the generator without relying on finite-dimensional assumptions or explicit spectral parameterizations. Experiments ranging from nonlinear oscillators to high-dimensional chaotic systems demonstrate that KoopGen substantially improves long-term prediction accuracy and stability, uncovering learnable and interpretable structural components underlying continuous-spectrum dynamics.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Representing and predicting high-dimensional and spatiotemporally chaotic dynamical systems remains a fundamental challenge in dynamical systems and machine learning. Although data-driven models can achieve accurate short-term forecasts, they often lack stability, interpretability, and scalability in regimes dominated by broadband or continuous spectra. Koopman-based approaches provide a principled linear perspective on nonlinear dynamics, but existing methods rely on restrictive finite-dimensional assumptions or explicit spectral parameterizations that degrade in high-dimensional settings. Against these issues, we introduce KoopGen, a generator-based neural Koopman framework that models dynamics through a structured, state-dependent representation of Koopman generators. By exploiting the intrinsic Cartesian decomposition into skew-adjoint and self-adjoint components, KoopGen separates conservative transport from irreversible dissipation while enforcing exact operator-theoretic constraints during learning. Across systems ranging from nonlinear oscillators to high-dimensional chaotic and spatiotemporal dynamics, KoopGen improves prediction accuracy and stability, while clarifying which components of continuous-spectrum dynamics admit interpretable and learnable representations.

Problem

• dynamical systems
• continuous spectra
• Koopman operator
• spatiotemporal chaos
• high-dimensional prediction

Innovation

• Koopman operator
• continuous spectra
• neural generator
• operator-theoretic constraints
• chaotic dynamics