Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems

The article presents a novel approach called equivariant flow matching, which integrates flow matching with equivariant architectures to model the probability distribution over bifurcation outcomes in nonlinear dynamical systems. By employing an optimal-transport-based coupling mechanism, the method effectively captures multimodal distributions and symmetry-breaking phenomena, demonstrating superior performance compared to traditional non-probabilistic and variational methods. This advancement is significant for practitioners as it provides a scalable framework for accurately modeling multistability in high-dimensional systems, enhancing the understanding and prediction of complex dynamical behaviors.