Nonlinear integro-differential operator regression with neural networks

While direct numerical simulation (DNS) may provide accurate data for the evolution of a fluid mechanical system, there are still many challenges in synthesizing information obtained from these simulations into reduced-order models. Machine learning has already shown promise as a tool for physical modeling. In this talk we discuss a technique for extracting nonlinear partial integro-differential equations from data using a combination of Fourier spectral methods and neural network regression. Using a database of DNS results for the fractional heat equation, the Burgers' equation, and the Kuramoto-Sivashinsky equation, we demonstrate that this technique is capable of recovering approximate equations. We also show that a subgrid scale model for the Burgers' equation can be recovered using filtered DNS results.