Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems
This addresses the problem of modeling multistability in high-dimensional systems for researchers in nonlinear dynamics and physics, offering a principled and scalable solution.
The paper tackled the problem of modeling multiple coexisting stable solutions in symmetry-breaking bifurcation problems, where deterministic machine learning models fail to capture this multiplicity. The result was that their flow matching method significantly outperformed non-probabilistic and variational methods in capturing multimodal distributions and symmetry-breaking bifurcations.
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.