Neural Conservation Laws: A Divergence-Free Perspective

Jack Richter-Powell, Yaron Lipman, Ricky T. Q. Chen

Keywords: density estimation, automatic differentiation, generative modeling, optimal transport maps, structured deep learning

Abstract: We investigate the parameterization of deep neural networks that by design satisfy the continuity equation, a fundamental conservation law. This is enabled by the observation that any solution of the continuity equation can be represented as a divergence-free vector field. We hence propose building divergence-free neural networks through the concept of differential forms, and with the aid of automatic differentiation, realize two practical constructions. As a result, we can parameterize pairs of densities and vector fields that always satisfy the continuity equation by construction, foregoing the need for extra penalty methods or expensive numerical simulation. Furthermore, we prove these models are universal and so can be used to represent any divergence-free vector field. Finally, we experimentally validate our approaches by computing neural network-based solutions to fluid equations, solving for the Hodge decomposition, and learning dynamical optimal transport maps.

TL;DR: We construct divergence-free neural networks and use these to parameterize solutions of the continuity equation, directly modeling pairs of density and vector fields that always satisfy the law of conservation.

Community Implementations: [ 6 code implementations](https://www.catalyzex.com/paper/arxiv:2210.01741/code)