Mechanistic PDE Networks for Discovery of Governing Equations

TL;DR

This paper introduces Mechanistic PDE Networks, a neural network framework that learns and solves governing PDEs from data, enabling discovery of complex dynamical systems with efficiency and robustness.

Contribution

The paper develops a GPU-capable multigrid PDE solver integrated into neural networks for discovering nonlinear PDEs from noisy data.

Findings

Successfully discovers reaction-diffusion PDEs.

Accurately models Navier-Stokes equations.

Robust to data noise and complex dynamics.

Abstract

We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in neural network hidden representations. The represented PDEs are then solved and decoded for specific tasks. The learned PDE representations naturally express the spatiotemporal dynamics in data in neural network hidden space, enabling increased power for dynamical modeling. Solving the PDE representations in a compute and memory-efficient way, however, is a significant challenge. We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations that acts as a module in Mechanistic PDE Networks. Leveraging the PDE solver, we propose a discovery architecture that can discover nonlinear PDEs in complex settings while also being robust to noise. We validate PDE discovery on a number of PDEs, including reaction-diffusion and Navier-Stokes equations.