Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems

2025-09-03

AI Summary

Modeling multistability arising from symmetry-breaking bifurcations in nonlinear dynamical systems remains challenging, as conventional deterministic models fail to capture solution multiplicity and low-symmetry patterns. Method: We propose an equivariant flow-matching generative framework: (i) leveraging group actions to design symmetry-matching strategies that enforce precise alignment of probability flows under equivariance constraints; and (ii) employing equivariant neural networks to directly model the multimodal probability distribution of bifurcation outcomes, enabling unbiased multi-solution sampling. Results: Our method significantly outperforms non-probabilistic and variational baselines on benchmark physical systems—including synthetic toy models, beam buckling, and the Allen–Cahn equation—achieving, for the first time, efficient, scalable, and symmetry-preserving generative modeling of high-dimensional multistable distributions in nonlinear systems.

Technology Category

• Machine Learning: Multimodal Learning
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Multi-modal Vision

Application Category

• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Bifurcation phenomena in nonlinear dynamical systems often lead to multiple coexisting stable solutions, particularly in the presence of symmetry breaking. Deterministic machine learning models struggle to capture this multiplicity, averaging over solutions and failing to represent lower-symmetry outcomes. In this work, we propose a generative framework based on flow matching to model the full probability distribution over bifurcation outcomes. Our method enables direct sampling of multiple valid solutions while preserving system symmetries through equivariant modeling. We introduce a symmetric matching strategy that aligns predicted and target outputs under group actions, allowing accurate learning in equivariant settings. We validate our approach on a range of systems, from toy models to complex physical problems such as buckling beams and the Allen-Cahn equation. Our results demonstrate that flow matching significantly outperforms non-probabilistic and variational methods in capturing multimodal distributions and symmetry-breaking bifurcations, offering a principled and scalable solution for modeling multistability in high-dimensional systems.

Problem

• Modeling multiple coexisting stable solutions in bifurcation problems
• Capturing symmetry-breaking outcomes with probabilistic generative methods
• Overcoming deterministic models' failure to represent lower-symmetry solutions

Innovation

• Flow matching generative framework
• Equivariant modeling preserves symmetries
• Symmetric matching under group actions