Improved Training of Strong Stochastic Flow Maps

Stochastic flow maps promise few-step sampling of stochastic differential equations (SDEs) by learning to directly jump between times along the solution path, yet training these models to learn the strong (pathwise) solution remains difficult. The primary obstacle is the Brownian path itself: it is an infinite dimensional object which needs to be passed to the neural network and is nowhere differentiable which makes the implementation of derivative-based losses which power the state-of-the-art deterministic flow maps just out of reach. We identify two core design principles which address these issues: a) represent the Brownian path with a basis which carries higher-order local information, and b) train with an objective that forces the network to use the full representation of the path not just the first order. To address this we leverage the shifted Legendre polynomial expansion with a Wong-Zakai styled objective unlocking derivative-based losses. Finally, we propose a recipe for fine-tuning large pre-trained flow models into strong stochastic flow maps. We ablate through this design space on several toy problems and use our findings to transform a large flow model into a strong stochastic flow map on ImageNet to achieve an FID-50k of 2.31 at 16 NFE.