Data-driven gradient flows

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 specialize 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 independently of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation using a 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 several numerical examples.