Semi-supervised learning of partial differential operators and dynamical flows

Michael Rotman, Amit Dekel, Ran Ilan Ber, Lior Wolf, Yaron Oz

Keywords: Hypernetworks, Partial Differential Equations, Fluid Dynamics

Abstract:

The evolution of dynamical systems is generically governed by nonlinear partial differential equations (PDEs), whose solution, in a simulation framework, requires vast amounts of computational resources. For a growing number of specific cases, neural network-based solvers have been shown to provide comparable results to other numerical methods while utilizing fewer resources.
In this work, we present a novel method that combines a hyper-network solver with a Fourier Neural Operator architecture. Our method treats time and space separately. As a result, it successfully propagates initial conditions in discrete time steps by employing the general composition properties of the partial differential operators. Following previous work, supervision is provided at a specific time point. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, and three spatial dimensions. The results show that the new method improves the learning accuracy at the time point of supervision point, and is also able to interpolate and extrapolate the solutions to arbitrary times.

One-sentence Summary: A novel architecture of solving PDEs in a semi-supervised manner.