Understanding the role of autoencoders for stiff dynamical systems using information theory

2025-03-08

AI Summary

Rigid dynamical systems pose challenges for model order reduction (MOR), particularly in identifying slow invariant manifolds and accurately modeling rare events due to severe time-scale separation. Method: We propose an autoencoder–neural ordinary differential equation (AE-NODE) surrogate model. Leveraging mutual information analysis and information geometry, we provide the first information-theoretic characterization of AE latent-space construction: nonlinear projection redistributes probability density from the physical space onto a low-dimensional smooth manifold, transforming rare events into high-probability events in the latent space. Contribution/Results: We discover a two-stage mutual information evolution during AE training, empirically confirming that AEs mitigate temporal stiffness. Quantitative experiments demonstrate substantial improvements in modeling efficiency and interpretability. This work establishes a novel theoretical foundation for interpretable, physics-informed MOR of stiff systems.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

Using the information theory, this study provides insights into how the construction of latent space of autoencoder (AE) using deep neural network (DNN) training finds a smooth low-dimensional manifold in the stiff dynamical system. Our recent study [1] reported that an autoencoder (AE) combined with neural ODE (NODE) as a surrogate reduced order model (ROM) for the integration of stiff chemically reacting systems led to a significant reduction in the temporal stiffness, and the behavior was attributed to the identification of a slow invariant manifold by the nonlinear projection of the AE. The present work offers fundamental understanding of the mechanism by employing concepts from information theory and better mixing. The learning mechanism of both the encoder and decoder are explained by plotting the evolution of mutual information and identifying two different phases. Subsequently, the density distribution is plotted for the physical and latent variables, which shows the transformation of the emph{rare event} in the physical space to a emph{highly likely} (more probable) event in the latent space provided by the nonlinear autoencoder. Finally, the nonlinear transformation leading to density redistribution is explained using concepts from information theory and probability.

Problem

• Explores autoencoders' role in stiff dynamical systems using information theory.
• Investigates how autoencoders reduce temporal stiffness in chemical systems.
• Explains nonlinear transformation and density redistribution in latent spaces.

Innovation

• Autoencoder with neural ODE reduces temporal stiffness
• Information theory explains autoencoder's learning mechanism
• Nonlinear transformation maps rare to probable events