Bi-FORK's inference-time particle-guidance sampler is presented by its authors as a parallelized analog of classical numerical deflation

physics-mlbifurcation-theorydeflationgenerative-modelingflow-matchingscientific-machine-learningarxivhistory-of-numerical-methods

At inference, rather than drawing K candidate trajectories independently, Bi-FORK integrates K of them through the same probability-flow ODE in a shared batch, adding an inter-sample repulsion force — weighted by a Gaussian RBF kernel and restricted to the predicted post-bifurcation frames — that pushes similar trajectories apart. The effect is that the sample set spreads across a bifurcating system's distinct physical branches rather than collapsing onto one.

The paper's authors frame this mechanism's lineage explicitly, writing that it acts "as a parallelized analog of classical deflation ... to recover distinct solutions." Deflation is a decades-old numerical technique for finding additional distinct solutions of a nonlinear system once one solution is already known, introduced for algebraic nonlinear systems by Kenneth M. Brown and William B. Gearhart (1971, Numerische Mathematik; Bi-FORK's own reference list misprints the first author as "Brow") and extended to nonlinear PDEs by Patrick Farrell, A. Birkisson and Simon Funke (2015, SIAM Journal on Scientific Computing). The repulsion mechanism itself is adapted from "particle guidance," a non-iid diverse-sampling technique for diffusion models introduced by Gabriele Corso and coauthors (2024, ICLR).

Part of the same reading as claim-bi-fork-whole-trajectory-latent-flow-matching-for-global-branch-consistency and claim-bi-fork-260k-point-scale-is-first-demonstration-of-one-to-many-bifurcation-recovery.