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

Methods of Branched Optimal Transport (BOT) mimic the economy and efficiency of natural tree-like structures, such as those found in rivers and biological systems. These methods are widely applicable for designing efficient networks in society, from river basins and blood vessels to mail and gas distribution systems. However, they remain understudied in the context of designing deep generative models, particularly at a large scale. Standard continuous-time generative models, such as the flow matching approach, fail to capture the inherent hierarchical and branching patterns present in real-world data. Current models provide no mechanism for flows to merge or share pathways to minimize total transport cost. Inspired by the "economy of scale" principle in BOT, we introduce a novel, scalable branched flow-matching algorithm designed to solve the branched optimal transport problem in high dimensions. Our method adapts the Benamou-Brenier continuous-time optimal transport formulation to learn branched generative flows. These flows allow probability mass to aggregate along common pathways before branching out to diverse targets. Parametrized by neural networks, our method effectively learns complex branched generative processes. We demonstrate its effectiveness on challenging high-dimensional tasks in biology and image generation.

Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary, discontinuous, and harder to learn than the underlying solution set. We study bifurcation models, a weight-tied dynamical view in which different initializations can converge to different stable equilibria, so the model represents an attractor landscape rather than one chosen branch. We prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics and that the induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular. Experiments on frustrated Ising models show that such dynamics can discover multiple valid equilibria without branch labels and outperform single-branch supervision. Allen--Cahn experiments further show that diversity is not automatic: it can be encouraged explicitly, but with an accuracy--diversity tradeoff.

Systematic out-of-distribution (OOD) generation remains a critical bottleneck for continuous-time generative models. While standard joint classifier-free guidance (CFG) routinely fails to synthesize unobserved concept combinations, exact decomposed scoring generalizes robustly at the cost of severe computational overhead. In this work, we reveal that compositional binding is not a uniform process but a highly localized phase transition. We identify the semantic bifurcation window - the precise temporal interval where joint and decomposed vector fields meaningfully diverge. Exploiting this dynamic, we propose surgical guidance, a hybrid sampling strategy that restricts exact multi-pass scoring strictly to this critical window. On an OOD bi-digit MNIST testbed, surgical guidance achieves state-of-the-art compositional fidelity at a fraction of the inference cost, yielding a +5.3% absolute improvement in pairwise accuracy over the joint baseline by intervening during just the first 15% of the diffusion trajectory. Furthermore, our empirical analysis uncovers a fundamental topological divide: diffusion models (SDEs) force conceptual resolution immediately at peak noise, whereas Conditional Flow Matching (ODEs) delays structural binding until intermediate features emerge, establishing a new temporal framework for accelerating large-scale generative decoding.