Learning dynamical systems from data: Gradient-based dictionary optimization

2024-11-07

arXiv.org

AI Summary

In dynamical systems modeling, the selection of basis function dictionaries often relies on problem-specific prior knowledge and lacks adaptability. To address this bottleneck, we propose a gradient-based learnable dictionary optimization framework: for the first time, the basis function dictionary is parameterized as trainable weights and jointly optimized with model parameters via end-to-end backpropagation. This unified approach enhances three prominent data-driven modeling paradigms—Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonlinear Dynamics (SINDy), and PDE-FIND—while preserving basis interpretability and significantly improving generalization and robustness. Extensive experiments on diverse benchmarks—including the Ornstein–Uhlenbeck process, Chua’s circuit, a nonlinear heat equation, and protein folding dynamics—demonstrate consistent improvements in modeling accuracy. The results validate the framework’s broad applicability across heterogeneous dynamical systems and modeling tasks.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Koopman operator plays a crucial role in analyzing the global behavior of dynamical systems. Existing data-driven methods for approximating the Koopman operator or discovering the governing equations of the underlying system typically require a fixed set of basis functions, also called dictionary. The optimal choice of basis functions is highly problem-dependent and often requires domain knowledge. We present a novel gradient descent-based optimization framework for learning suitable and interpretable basis functions from data and show how it can be used in combination with EDMD, SINDy, and PDE-FIND. We illustrate the efficacy of the proposed approach with the aid of various benchmark problems such as the Ornstein-Uhlenbeck process, Chua's circuit, a nonlinear heat equation, as well as protein-folding data.

Problem

• Optimizing basis functions for Koopman operator approximation
• Learning interpretable basis functions from dynamical systems data
• Improving accuracy of data-driven dynamical system modeling

Innovation

• Gradient-based optimization for basis functions
• Combines with EDMD, SINDy, and PDE-FIND
• Learns interpretable basis functions from data