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

arXiv: https://arxiv.org/abs/2610.12449 · HTML: https://mlsanity.com/paper/2610.12449/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 61 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 7 of 7)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [CRPS-LAM: Probabilistic Regional Weather Forecasting with Continuous Ranked Probability Score](https://mlsanity.com/paper/2510.09484/) (2025)
- [Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems](https://mlsanity.com/paper/2509.03340/) (2025)
- [Skillful High-Resolution Ensemble Precipitation Forecasting with an Integrated Deep Learning Framework](https://mlsanity.com/paper/2501.02905/) (2025)
- [Multimodal Scientific Learning Beyond Diffusions and Flows](https://mlsanity.com/paper/2602.00960/) (2026)
- [Learning Structure, Energy, and Dynamics: A Survey of Artificial Intelligence for Protein Dynamics](https://mlsanity.com/paper/2604.25244/) (2026)

## Abstract (arXiv)

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.