# Fisher information regularization schemes for Wasserstein gradient flows

## TL;DR

This paper introduces a Fisher information regularized variational scheme for Wasserstein gradient flows, improving convexity, stability, and computational efficiency, with applications to various PDEs.

## Contribution

It develops a novel regularization approach based on Fisher information within the Jordan--Kinderlehrer--Otto framework, enhancing numerical stability and efficiency.

## Key findings

- Improves convexity and stability of Wasserstein gradient flow computations.
- Reduces computational cost by eliminating the need for additional time interpolation.
- Demonstrates effectiveness on multiple PDE examples, including porous media and nonlinear Fokker-Planck equations.

## Abstract

We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan--Kinderlehrer--Otto framework with the Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and adds a regularization by the Fisher information. This regularization can be derived in terms of energy splitting and is closely related to the Schr{\"o}dinger bridge problem. It improves the convexity of the variational problem and automatically preserves the non-negativity of the solution. As a result, it allows us to apply sequential quadratic programming to solve the sub-optimization problem. We further save the computational cost by showing that no additional time interpolation is needed in the underlying dynamic formulation of the Wasserstein-2 metric, and therefore, the dimension of the problem is vastly reduced. Several numerical examples, including porous media equation, nonlinear Fokker-Planck equation, aggregation diffusion equation, and Derrida-Lebowitz-Speer-Spohn equation, are provided. These examples demonstrate the simplicity and stableness of the proposed scheme.