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This research introduces a generative AI framework using flow matching and equivariant architectures to accurately model symmetry-breaking bifurcations in nonlinear dynamical systems. By capturing multimodal probability distributions where deterministic models fail, the method offers a scalable solution for analyzing multistability in high-dimensional physical problems like buckling beams.
Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems
arXiv:2509.03340v4 Announce Type: replace-cross 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 are unable to capture this multiplicity, averaging over solutions and failing to represent lower-symmetry outcomes. In this work, we formalize the use of generative AI, specifically flow matching, as a principled way to model the full probability distribution over bifurcation outcomes. Our approach builds on existing techniques by combining flow matching with equivariant architectures and an optimal-transport-based coupling mechanism. We generalize equivariant flow matching to a symmetric coupling 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 simple conceptual systems to physical problems such as buckling beams and the Allen--Cahn equation. The results demonstrate that the approach accurately captures multimodal distributions and symmetry-breaking bifurcations. Moreover, our results demonstrate that flow matching significantly outperforms non-probabilistic and variational methods. This offers a principled and scalable solution for modeling multistability in high-dimensional 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 are unable to capture this multiplicity, averaging over solutions and failing to represent lower-symmetry outcomes. In this work, we formalize the use of generative AI, specifically flow matching, as a principled way to model the full probability distribution over bifurcation outcomes. Our approach builds on existing techniques by combining flow matching with equivariant architectures and an optimal-transport-based coupling mechanism. We generalize equivariant flow matching to a symmetric coupling 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 simple conceptual systems to physical problems such as buckling beams and the Allen--Cahn equation. The results demonstrate that the approach accurately captures multimodal distributions and symmetry-breaking bifurcations. Moreover, our results demonstrate that flow matching significantly outperforms non-probabilistic and variational methods. This offers a principled and scalable solution for modeling multistability in high-dimensional systems.
9 pages, 7 figures including appendices. Accepted to Machine Learning and the Physical Sciences Workshop, NeurIPS 2025 (this https URL). Repository with corresponding code: this https URL. Video explanation: this https URL
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE); Computational Physics (physics.comp-ph)
arXiv:2509.03340 [cs.LG]
(or arXiv:2509.03340v4 [cs.LG] for this version)
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