Physics-informed reduced-order modelling with equivariant spectral submanifolds

2026-08-04

AI Summary

This work addresses the high computational cost and poor robustness commonly encountered in spectral submanifold (SSM)-based reduced-order modeling of high-dimensional nonlinear systems. It reveals for the first time that SSMs inherently possess equivariance properties and leverages this insight to propose an equivariant spectral submanifold (eSSM) framework. By explicitly embedding the symmetries of the full-order model, eSSM integrates physical priors with group-action structures to construct an efficient nonlinear dimensionality reduction approach. Grounded in group representation theory, the method introduces equivariant manifold modeling and an associated SSM reduction algorithm. Numerical experiments across multiple benchmark problems—including tests within the Scientific Machine Learning Universal Tasks framework—demonstrate significant improvements in computational efficiency, accuracy, and numerical stability compared to conventional approaches.

Abstract

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

Problem

• spectral submanifold
• reduced-order modeling
• equivariance
• symmetry
• computational efficiency

Innovation

• equivariant reduction
• spectral submanifold
• symmetry
• reduced-order modeling
• nonlinear dynamics