Data driven gradient flows

TL;DR

This paper introduces a flexible variational framework for gradient flows in metric spaces, enabling the treatment of various PDEs through a unified approach and a simple change of the driving functional.

Contribution

It develops a general variational data assimilation method for gradient flows in metric spaces, with a focus on Wasserstein spaces, and provides a versatile numerical implementation.

Findings

Applicable to many nonlinear PDEs like porous medium and drift-diffusion equations

Uses a primal-dual algorithm for efficient numerical solutions

Demonstrates effectiveness through detailed numerical examples

Abstract

We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan-Kinderlehrer-Otto) approximation scheme. After discussing stability properties in the most general case, we specialise to the space of probability measures endowed with the Wasserstein distance. This setting covers many non-linear partial differential equations (PDEs), such as the porous medium equation or general drift-diffusion-aggregation equations, which can be treated by our methods independent of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation of our approach using an primal-dual algorithm. The strength of our approach lies in the fact that by simply changing the driving functional, a wide range of PDEs can be treated without the need to adopt the numerical scheme. We conclude by presenting detailed numerical examples.