Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems

TL;DR

This work tackles the challenge of modeling multistability and symmetry-breaking in nonlinear dynamical systems by combining equivariant modeling with flow matching to learn the full probability distribution over bifurcation outcomes. By enforcing symmetry via stabilizer subgroups and introducing symmetric coupling to align equivalent solutions, the method can sample multiple valid outcome trajectories rather than collapsing to a mean. Empirical results on toy problems, buckling beams, and the Allen–Cahn equation demonstrate superior performance over non-probabilistic and variational approaches in capturing multimodal and symmetry-broken behavior, including pitchfork bifurcations. The approach provides a principled, scalable framework for symmetry-aware generative modeling of high-dimensional dynamical systems with multistability and symmetry-breaking.

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.

Figures (7)

• Figure 1: (a) Self-similarity and equivariance illustrated with the buckling beam problem, in which the input is the state at the current time step $t$ and the target output is the next time step $t+1$. $d$ is the relative vertical downward displacement of the beam tip, which is increased over time. The relevant symmetry group $G$ here consists of only two elements: identity and reflection in the y-axis. In this case, the self-similarity of the beam before buckling happens to be described by the same group, i.e., here $G_x=G$. (b) The $x$-coordinates of each node over the course of a trajectory, showing 50 predictions of a trained flow matching model (dashed lines) compared to the two possible ground truths (solid lines). (c) The same 50 predictions compared to the two possible ground truths, showing the beam deformation.
• Figure 2: Illustration of the 3 roads problem. (a) Input, (b) possible outputs, (c) what that looks like in the actual data.
• Figure 3: Illustration of the 4 node graph problem.
• Figure 4: Top: the results of predicting a probability distribution consisting of two Dirac deltas, using a VAE and flow matching. Bottom: the learned flow field by the trained flow matching model (contour plot), showing how samples from the Gaussian prior are pushed towards the two delta peaks (black lines).
• Figure 5: The results of predicting outcomes of a coin flip using three different methods: (a) a regular neural network, (b) a conditional VAE, and (c) flow matching. Each method makes 10 predictions per test data point.