Learning zeros of Fokker-Planck operators

TL;DR

This paper introduces a deep learning algorithm to find zeros of Fokker-Planck operators with non-solenoidal drift, demonstrating scalability and potential advantages over Monte Carlo methods in various dimensions.

Contribution

The paper presents a novel deep learning approach for solving Fokker-Planck equations, capable of handling higher dimensions and providing functional solutions, outperforming traditional Monte Carlo methods in accuracy.

Findings

Algorithm scales linearly in memory with dimension

Computational time scales quadratically with dimension

Deep network solutions can surpass Monte Carlo accuracy

Abstract

In this paper we devise a deep learning algorithm to find non-trivial zeros of Fokker-Planck operators when the drift is non-solenoidal. We demonstrate the efficacy of our algorithm for problem dimensions ranging from 2 to 10. This method scales linearly with dimension in memory usage. In the problems we studied, overall computational time seems to scale approximately quadratically with dimension. We present results that indicate the potential of this method to produce better approximations compared to Monte Carlo methods, for the same overall sample sizes, even in low dimensions. Unlike the Monte Carlo methods, the deep network method gives a functional form of the solution. We also demonstrate that the associated loss function is strongly correlated with the distance from the true solution, thus providing a strong numerical justification for the algorithm. Moreover, this relation seems to be linear asymptotically for small values of the loss function.