SINDy on slow manifolds

2025-07-01

AI Summary

For high-dimensional slow–fast systems, the Sparse Identification of Nonlinear Dynamics (SINDy) method faces computational intractability and severe ill-conditioning due to exponentially growing library size. To address this, we propose a two-stage manifold-aware modeling framework: first, data-driven learning of algebraic constraints relating fast variables to slow variables to explicitly construct the slow manifold; second, derivation of a physically consistent, sparse function library—retaining only essential higher-order nonlinear terms—based on the identified manifold. This drastically reduces library dimensionality and improves numerical conditioning. Crucially, the method jointly identifies the geometric structure of the slow manifold and the associated reduced dynamics, circumventing the “library explosion” caused by conventional truncation strategies. Numerical experiments on beam snap-through buckling and flow past a NACA 0012 airfoil demonstrate that the approach successfully recovers high-fidelity reduced-order models of slow-manifold dynamics, achieving one- to two-order-of-magnitude reduction in library size and substantial mitigation of ill-conditioning.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The sparse identification of nonlinear dynamics (SINDy) has been established as an effective method to learn interpretable models of dynamical systems from data. However, for high-dimensional slow-fast dynamical systems, the regression problem becomes simultaneously computationally intractable and ill-conditioned. Although, in principle, modeling only the dynamics evolving on the underlying slow manifold addresses both of these challenges, the truncated fast variables have to be compensated by including higher-order nonlinearities as candidate terms for the model, leading to an explosive growth in the size of the SINDy library. In this work, we develop a SINDy variant that is able to robustly and efficiently identify slow-fast dynamics in two steps: (i) identify the slow manifold, that is, an algebraic equation for the fast variables as functions of the slow ones, and (ii) learn a model for the dynamics of the slow variables restricted to the manifold. Critically, the equation learned in (i) is leveraged to build a manifold-informed function library for (ii) that contains only essential higher-order nonlinearites as candidate terms. Rather than containing all monomials of up to a certain degree, the resulting custom library is a sparse subset of the latter that is tailored to the specific problem at hand. The approach is demonstrated on numerical examples of a snap-through buckling beam and the flow over a NACA 0012 airfoil. We find that our method significantly reduces both the condition number and the size of the SINDy library, thus enabling accurate identification of the dynamics on slow manifolds.

Problem

• Improves SINDy for high-dimensional slow-fast systems
• Identifies slow manifold and dynamics efficiently
• Reduces library size and condition number

Innovation

• Two-step SINDy for slow-fast dynamics
• Manifold-informed sparse custom library
• Reduced condition number and library size