Mechanistic PDE Networks for Discovery of Governing Equations
This work addresses the challenge of PDE discovery for researchers in computational science and machine learning, offering a method that is robust to noise and applicable to complex settings, though it appears incremental as it builds on existing neural network and solver techniques.
The authors tackled the problem of discovering governing partial differential equations from data by introducing Mechanistic PDE Networks, which represent spatiotemporal dynamics as linear PDEs in neural network hidden spaces and use a specialized multigrid solver for efficient computation, achieving validation on equations like reaction-diffusion and Navier-Stokes.
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.