Deep Invertible Autoencoders for Dimensionality Reduction of Dynamical Systems

2026-03-13

AI Summary

This work proposes a deep invertible autoencoder (inv-AE) to address challenges in traditional reduced-order modeling of convection-dominated problems, such as slow singular value decay and projection errors that stagnate in plateaus. By introducing invertible neural networks into dynamical system dimensionality reduction for the first time, the method employs a multi-layer invertible architecture that progressively recovers full-order information as the manifold dimension increases, effectively mitigating error plateauing. Numerical experiments on a parametrized 1D Burgers equation and a 2D flow past variable-geometry bodies demonstrate that inv-AE significantly reduces projection error, enhances reconstruction accuracy, and can be seamlessly integrated with mainstream projection-based reduced-order modeling techniques to improve overall performance.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of high-dimensional, parametric systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose a deep invertible AE architecture, named inv-AE, that improves upon the stagnation of the projection error typical of traditional AE architectures, e.g., convolutional, and the reconstructions quality. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a parametric 1D Burgers' equation and a parametric 2D fluid flow around an obstacle with variable geometry, we show that (i) inv-AE mitigates the issue of the characteristic plateau of (convolutional) AEs and (ii) inv-AE can be combined with popular projection-based ROM approaches to improve their accuracy.

Problem

• dimensionality reduction
• reduced-order modeling
• autoencoders
• projection error plateau
• dynamical systems

Innovation

• invertible autoencoder
• dimensionality reduction
• reduced-order modeling
• deep learning
• dynamical systems