The inference of Fokker-Planck equations via transport maps

TL;DR

This paper introduces a Bayesian framework that reconstructs Fokker-Planck equations from population trajectories using transport maps and variational inference, enabling efficient inference in multi-dimensional spaces.

Contribution

It presents a novel method combining transport maps and Bayesian inference to infer Fokker-Planck equations directly from trajectory data, handling uncertainty and high-dimensional spaces.

Findings

Accurately infers Fokker-Planck dynamics from trajectory data.

Efficiently handles high-dimensional spaces up to five dimensions.

Demonstrates robustness with limited data.

Abstract

We present a framework, which, from the trajectories detailing the spatiotemporal dynamics of a population, simultaneously reconstructs a transport map as well as the Fokker-Planck equation governing the coarse-grained probability distribution. Leveraging the Knothe-Rosenblatt rearrangement, we model the transport map from a fixed reference distribution to the target distribution, and derive the velocity fields of the flows from the trajectory of transport maps. Exploiting the velocity fields, we circumvent spatial gradients to infer the Fokker-Planck equation's potential and diffusivity. The sparsity of trajectories injects uncertainty, which we treat in a Bayesian setting using variational inference. The approach is applied to inferring the Fokker-Planck dynamics in spaces of up to five dimensions, demonstrating both accurate identification of the system and efficiency with respect to data size.