# Gradient Flow Algorithms for Density Propagation in Stochastic Systems

## TL;DR

This paper introduces a novel gradient flow-based computational framework for propagating joint probability density functions in stochastic nonlinear systems, avoiding traditional discretization limitations and enabling fast, non-parametric uncertainty analysis.

## Contribution

It develops a time-discretized, geometry-inspired method that solves infinite-dimensional PDEs via proximal recursions on the space of PDFs, with guaranteed convergence and efficiency.

## Key findings

- Achieves fast, non-parametric joint PDF propagation.
- Proves contractiveness of the fixed point recursion.
- Demonstrates effectiveness through numerical examples.

## Abstract

We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the transient joint PDFs subject to prior dynamics arises in uncertainty propagation, nonlinear filtering and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization or function approximation -- both of which, in general, suffer from the "curse-of-dimensionality". In the proposed framework, we discretize time but not the state space. We solve infinite dimensional proximal recursions in the manifold of joint PDFs, which in the small time-step limit, is theoretically equivalent to solving the underlying transport PDEs. The resulting computation has the geometric interpretation of gradient flow of certain free energy functional with respect to the Wasserstein metric arising from the theory of optimal mass transport. We show that dualization along with an entropic regularization, leads to a cone-preserving fixed point recursion that is proved to be contractive in Thompson metric. A block co-ordinate iteration scheme is proposed to solve the resulting nonlinear recursions with guaranteed convergence. This approach enables remarkably fast computation for non-parametric transient joint PDF propagation. Numerical examples and various extensions are provided to illustrate the scope and efficacy of the proposed approach.