Learning the Dynamics of Chaos and Turbulence

The evolution of many dynamical systems is governed by nonlinear partial differential equations (PDEs), whose solution, in a simulation framework, requires vast amounts of computational resources.

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. 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.

In the figure we see the velocity vector fields, and the corresponding vorticity of the two-dimensional Burgers equation. Shock waves are being formed in the final state and marked by the zoom- ins.