Bi-FORK: Generative Modeling of High-Dimensional Bifurcating Systems

2026-10-08

AI Summary

This study addresses the one-to-many mapping challenge arising from bifurcations in high-dimensional physical systems, overcoming the single-solution assumption inherent in conventional deep learning by proposing a novel generative solving framework. Methodologically, this work pioneers the application of generative modeling to high-dimensional bifurcation systems, employing latent flow matching to ensure spatiotemporal coherence and integrating repulsive guidance sampling to recover multiple branches under symmetry breaking. Experimental results demonstrate that the proposed method successfully reconstructs multimodal solution structures across tasks including buckling beams, mechanical metamaterials, and phase separation. A single amortized inference pass suffices to capture all solution branches, with discretization scaling up to 260,000 points, significantly outperforming existing baseline methods.

Abstract

Bifurcations are ubiquitous in physical systems, from structural buckling to fluid and climate dynamics, yet they remain largely unexplored in deep learning. At a symmetry-breaking bifurcation, a single input admits multiple equally valid solutions, violating the one-to-one assumption underlying most learned physical surrogates. We introduce Bi-FORK, a generative framework for learning these one-to-many solution maps in high-dimensional systems. Bi-FORK generates complete trajectories through latent flow matching, preserving space and time coherence, and uses repulsion-guided sampling to recover distinct solution branches in a single amortized pass. We evaluate Bi-FORK on buckling beams, mechanical metamaterials, and Allen-Cahn phase separation, spanning continuous, discrete, and field-valued bifurcations with discretizations up to 260,000 points. Bi-FORK recovers the multimodal solution structure while scaling several orders of magnitude beyond prior approaches, opening generative modeling to high-dimensional bifurcating physical systems.

Problem

• bifurcation
• generative modeling
• one-to-many mapping
• high-dimensional systems
• symmetry-breaking

Innovation

• generative modeling
• bifurcation
• flow matching
• repulsion-guided sampling
• high-dimensional systems