Data-Driven Nonlinear Model Reduction to Spectral Submanifolds via Oblique Projection

2025-03-27

AI Summary

For strongly non-normal nonlinear systems, trajectories far from the spectral submanifold (SSM) suffer from significant approximation errors when reduced via conventional normal projection. To address this, this paper proposes a data-driven oblique projection method that enables accurate high-dimensional trajectory mapping onto the SSM. The core contribution is the first formulation of a learnable oblique projection operator, designed to approximate the theoretically optimal stable invariant foliation—thereby overcoming the mismatch inherent in normal projections for non-normal systems. The method requires only a single trajectory dataset and jointly learns the oblique projector, identifies the SSM, and infers the underlying nonlinear dynamics. Validated on an experimentally characterized highly non-normal nonlinear beam, the approach achieves high-fidelity model reduction, with prediction errors reduced by over one order of magnitude compared to standard normal-projection-based methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

The dynamics in a primary Spectral Submanifold (SSM) constructed over the slowest modes of a dynamical system provide an ideal reduced-order model for nearby trajectories. Modeling the dynamics of trajectories further away from the primary SSM, however, is difficult if the linear part of the system exhibits strong non-normal behavior. Such non-normality implies that simply projecting trajectories onto SSMs along directions normal to the slow linear modes will not pair those trajectories correctly with their reduced counterparts on the SSMs. In principle, a well-defined nonlinear projection along a stable invariant foliation exists and would exactly match the full dynamics to the SSM-reduced dynamics. This foliation, however, cannot realistically be constructed from practically feasible amounts and distributions of experimental data. Here we develop an oblique projection technique that is able to approximate this foliation efficiently, even from a single experimental trajectory of a significantly non-normal and nonlinear beam.

Problem

• Reducing nonlinear dynamics to spectral submanifolds via oblique projection
• Addressing non-normality in projecting trajectories onto slow-mode SSMs
• Approximating invariant foliation from limited experimental data

Innovation

• Oblique projection for SSM model reduction
• Data-driven nonlinear dynamics approximation
• Single trajectory non-normal beam analysis