Generative modeling of high-dimensional bifurcating systems



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.



From one-to-one to one-to-many



Deep learning surrogates for physical systems have largely operated on the assumption that each system state maps to a unique successor state. This one-to-one constraint simplifies training but breaks down at symmetry-breaking bifurcations, where multiple distinct futures are equally possible from a given configuration. Structural buckling of a beam under load, phase separation in a binary fluid, or patterning in a reaction-diffusion system all exhibit this multiplicity: the system can evolve along different solution branches depending on infinitesimal perturbations that cannot be predicted from the governing equations alone.



Bi-FORK architecture



The core innovation of Bi-FORK is a generative model that does not attempt to predict a single next state but instead learns the full solution manifold. The framework operates in latent space using flow matching, where trajectories are generated by interpolating between a noise distribution and the data distribution of system evolutions. This latent flow matching preserves space and time coherence, meaning that generated trajectories respect both the spatial geometry of the system and the temporal ordering of states.



Recovering distinct solution branches in a single amortized pass is the critical technical contribution. Prior work on multi-modal physical modeling either required separate passes per branch, exponential sampling, or post-hoc clustering that discarded temporal coherence. Bi-FORK uses repulsion-guided sampling, which explicitly pushes generated trajectories apart in latent space, ensuring that multiple modes are captured without requiring multiple forward passes or separate models per branch.



Evaluation across bifurcation types



The authors evaluate Bi-FORK on three diverse problem domains:



    
  
  • Buckling beams: continuous bifurcation where a straight beam buckles under increasing load, admitting symmetric left-leaning and right-leaning configurations.
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  • Mechanical metamaterials: discrete bifurcation involving multiple stable configurations of a micro-structured material under deformation.
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  • Allen-Cahn phase separation: field-valued bifurcation describing how an order parameter evolves from a uniform state into spatial patterns of coexisting phases.
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These domains span continuous and discrete bifurcations, as well as field-valued systems where the solution is a function defined over a spatial grid. The discretizations reach up to 260,000 points, placing the work in the high-dimensional regime where most generative models struggle.



Scaling beyond prior approaches



Bi-FORK recovers the multimodal solution structure while scaling several orders of magnitude beyond prior approaches. The paper reports results comparing against baseline methods—including conditional normalizing flows, GAN-based generators, and diffusion models adapted to physical systems—showing that Bi-FORK captures more solution branches with higher fidelity and greater temporal coherence. The scaling behavior is a key advantage: as the number of spatial discretization points increases, Bi-FORK's performance degrades gracefully while prior approaches fail entirely or require prohibitive computation.



Opening generative modeling to bifurcating systems



The broader impact of the work is to open generative modeling to a class of physical systems that have been largely absent from the deep learning literature. Bifurcating systems appear everywhere: in structural engineering, fluid dynamics, climate modeling, materials science, and biological pattern formation. By providing a framework that can generate complete, coherent solution branches at scale, Bi-FORK enables new possibilities for design exploration, uncertainty quantification, and scenario planning in any domain where multiple valid futures are possible from a given starting state.



Limitations and future directions



The authors acknowledge that repulsion-guided sampling may require careful tuning of the repulsion strength relative to the flow matching objective, and that extremely high-dimensional systems with many coexisting branches may still pose challenges. Extending the framework to bifurcations with continuous symmetry groups (rather than discrete modes) and to systems with stochastic forcing are natural directions for future work.



Conclusion



Bi-FORK introduces a generative framework for learning one-to-many solution maps in high-dimensional bifurcating systems. By generating complete trajectories through latent flow matching and using repulsion-guided sampling to recover distinct solution branches in a single amortized pass, the framework preserves space and time coherence while scaling to discretizations of 260,000 points. Evaluated on buckling beams, mechanical metamaterials, and Allen-Cahn phase separation, Bi-FORK recovers the multimodal solution structure several orders of magnitude beyond prior approaches, opening generative modeling to high-dimensional bifurcating physical systems. Read the paper on arXiv