Taming High-Dimensional Dynamics: Learning Optimal Projections onto Spectral Submanifolds

2025-04-04

AI Summary

Modeling and control of high-dimensional nonlinear systems face challenges including substantial model reduction errors and geometric structure mismatches. This paper proposes a fiber-aligned optimal projection method tailored for spectral submanifolds (SSMs), which rigorously derives a non-orthogonal optimal projection satisfying differential-geometric constraints—overcoming the structural limitations of conventional orthogonal projections on nonlinear manifolds—and enabling embedding of controllable systems. The method integrates data-driven SSM learning, geometry-constrained optimization, nonlinear dimensionality reduction, and model predictive control. Evaluated on a 180-dimensional robotic system, it achieves up to a fivefold improvement in trajectory tracking accuracy over state-of-the-art methods, while significantly enhancing long-term prediction fidelity and closed-loop control performance.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Non-convex Optimization

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

High-dimensional nonlinear systems pose considerable challenges for modeling and control across many domains, from fluid mechanics to advanced robotics. Such systems are typically approximated with reduced order models, which often rely on orthogonal projections, a simplification that may lead to large prediction errors. In this work, we derive optimality of fiber-aligned projections onto spectral submanifolds, preserving the nonlinear geometric structure and minimizing long-term prediction error. We propose a computationally tractable procedure to approximate these projections from data, and show how the effect of control can be incorporated. For a 180-dimensional robotic system, we demonstrate that our reduced-order models outperform previous state-of-the-art approaches by up to fivefold in trajectory tracking accuracy under model predictive control.

Problem

• High-dimensional nonlinear systems modeling challenges
• Optimal projections onto spectral submanifolds derivation
• Reduced-order models accuracy improvement demonstration

Innovation

• Fiber-aligned projections onto spectral submanifolds
• Computationally tractable data-driven approximation
• Enhanced trajectory tracking in model predictive control