Mechanistic PDE Networks for Discovery of Governing Equations

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 modeling power. 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 discovers nonlinear PDEs in complex settings, while being robust to noise. We validate PDE discovery on a number of PDEs including reaction-diffusion and Navier-Stokes equations.

Scientists use observations of natural phenomena to develop general laws that describe the observed phenomena. This process is manual, time consuming and increasing difficult as the amount of available data increases at a fast rate. Machine learning is a tool that can help scientists manage the vast amounts of data. We develop a method for discovery of natural laws directly from data, helping scientists to quickly develop and test theories. We develop our method by combining the structure of scientific laws and data driven methods. The resulting techniques allow scientists to test more complex data and represent more complex equations than prior methods.