A dynamical systems 'basis' for vortex interactions via differentiable programming and machine learning

在流体动力学中，湍流的动力系统观点认为湍流流体流动是一个非常高维的动力系统中的轨迹。该轨迹紧密地遮蔽了该动力系统的不稳定周期轨道。我们的目标是调查这些不稳定的周期轨道，并计算它们之间存在的任何异宿连接，使用新的计算工具，如微分规划。众所周知，欧拉方程（Navier-Stokes方程的无粘极限）的任何光滑连续解都可以通过取点涡的适当空间分布的极限来获得。为此，我们将这些新的计算工具应用于点涡模型，并发现了一个密集的不变解库。我们还旨在发现，如果有的话，这些不变的点涡的解决方案是动态相关的粘性湍流。

The dynamical systems view of turbulence in fluid dynamics considers turbulent fluid flow as a trajectory in a very high-dimensional dynamical system. The trajectory shadows the unstable periodic orbits of this dynamical system closely. We aim to investigate these unstable periodic orbits, and compute any heteroclinic connections which exist between them, using new computational tools such as differentiable programming. It is also well known that any smoothly continuous solution of the Euler equation (the inviscid limit of the Navier-Stokes equations) can be obtained by taking the limit of suitable spatial distributions of point vortices. For this reason, we apply these new computational tools to the point vortex model, and discover a dense library of invariant solutions. We also aim to discover which, if any, of these invariant point vortex solutions are dynamically relevant in viscous turbulent flow.