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.

翻译：非线性动力系统中的分岔现象常导致多个稳定解共存，尤其在对称破缺情形下。确定性机器学习模型无法捕捉这种多重性，其预测会平均不同解并无法表示低对称性结果。本文正式提出使用生成式人工智能（特别是流匹配）作为建模分岔结果完整概率分布的原理性方法。我们的方法基于现有技术，将流匹配与等变架构及基于最优传输的耦合机制相结合。我们将等变流匹配推广至对称耦合策略，该策略在群作用下对齐预测输出与目标输出，从而实现在等变设定下的精确学习。我们在从简单概念系统到物理问题（如屈曲梁和Allen-Cahn方程）的一系列系统中验证了该方法。结果表明，该方法能准确捕捉多峰分布与对称破缺分岔现象。此外，我们的结果证明流匹配显著优于非概率方法与变分方法。这为高维系统中的多稳态建模提供了原理性且可扩展的解决方案。