Leveraging time and parameters for nonlinear model reduction methods

2025-01-07

AI Summary

Traditional nonlinear model order reduction (MOR) methods struggle to simultaneously achieve high accuracy and efficiency for wave propagation and transport-dominated problems, due to slow decay of the Kolmogorov $n$-width. Method: This paper proposes a novel MOR framework based on temporal-domain expansion and linear encoding. It leverages joint expansion of system dynamics and temporal information to replace the nonlinear encoder in autoencoders with a simple linear projection—enabling high-fidelity state compression without nonlinearity. Contribution/Results: We provide the first theoretical proof that such linear encoding suffices under temporal expansion. The method reduces the number of tunable hyperparameters by ~50%, effectively mitigating the Kolmogorov barrier. It significantly lowers training time and computational cost while preserving approximation accuracy. Experiments demonstrate strong generalization, ease of training, and good scalability—establishing a practical new paradigm for real-time simulation and control of high-dimensional nonlinear dynamical systems.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

In this paper, we consider model order reduction (MOR) methods for problems with slowly decaying Kolmogorov $n$-widths as, e.g., certain wave-like or transport-dominated problems. To overcome this Kolmogorov barrier within MOR, nonlinear projections are used, which are often realized numerically using autoencoders. These autoencoders generally consist of a nonlinear encoder and a nonlinear decoder and involve costly training of the hyperparameters to obtain a good approximation quality of the reduced system. To facilitate the training process, we show that extending the to-be-reduced system and its corresponding training data makes it possible to replace the nonlinear encoder with a linear encoder without sacrificing accuracy, thus roughly halving the number of hyperparameters to be trained.

Problem

• Kolmogorov n-width
• model complexity
• autoencoder training

Innovation

• Model Order Reduction
• Linear Encoder
• Kolmogorov n-width