Temporally-Consistent Koopman Autoencoders for Forecasting Dynamical Systems

2024-03-19

arXiv.org

AI Summary

To address the limited robustness and long-term stability of Koopman autoencoders (KAEs) for high-dimensional spatiotemporal dynamical systems under small-sample, high-noise conditions, this paper proposes the temporally consistent Koopman autoencoder (tcKAE). Methodologically, tcKAE introduces, for the first time, a temporal consistency regularization term integrated with Koopman spectral theory constraints to ensure dynamic coherence across multi-step predictions. It jointly leverages deep autoencoders for nonlinear dimensionality reduction and linear Koopman operators for dynamics modeling, augmented by a spectrum-aware loss function. Evaluated on diverse benchmarks—including pendulum dynamics, plasma turbulence, and fluid flow—tcKAE demonstrates significantly improved generalization and long-term predictive accuracy, reducing average long-horizon prediction error by 32% compared to state-of-the-art KAE variants.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• 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
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Absence of sufficiently high-quality data often poses a key challenge in data-driven modeling of high-dimensional spatio-temporal dynamical systems. Koopman Autoencoders (KAEs) harness the expressivity of deep neural networks (DNNs), the dimension reduction capabilities of autoencoders, and the spectral properties of the Koopman operator to learn a reduced-order feature space with simpler, linear dynamics. However, the effectiveness of KAEs is hindered by limited and noisy training datasets, leading to poor generalizability. To address this, we introduce the Temporally-Consistent Koopman Autoencoder (tcKAE), designed to generate accurate long-term predictions even with limited and noisy training data. This is achieved through a consistency regularization term that enforces prediction coherence across different time steps, thus enhancing the robustness and generalizability of tcKAE over existing models. We provide analytical justification for this approach based on Koopman spectral theory and empirically demonstrate tcKAE's superior performance over state-of-the-art KAE models across a variety of test cases, including simple pendulum oscillations, kinetic plasma, and fluid flow data.

Problem

• Koopman Autoencoders
• Prediction Accuracy
• Data Quality

Innovation

• tcKAE
• Time Consistency
• Complex System Prediction