Semi-supervised Learning of Partial Differential Operators and Dynamical Flows

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

Abstract: The evolution of many dynamical systems is generically governed by nonlinear partial differential equations (PDEs), whose solution, in a simulation framework, requires vast amounts of computational 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 and as a result, it successfully propagates initial conditions in continuous time steps by employing the general composition properties of the partial differential operators. Following previous works, supervision is provided at a specific time point. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, or three spatial dimensions. The results show that the new method improves the learning accuracy at the time of the supervision point, and can interpolate the solutions to any intermediate time. Our implementation is available at \href{https://github.com/rotmanmi/Semi-Supervised-Learning-of-Dynamical-Flows}{https://github.com/rotmanmi/Semi-Supervised-Learning-of-Dynamical-Flows}.