Our understanding and ability to compute the solutions to nonlinear partial differential equations has been strongly curtailed by our inability to effectively parameterize the inertial manifold of their solutions. I will discuss our ongoing efforts for using machine learning to advance the state of the art, both for developing a qualitative understanding of "turbulent" solutions and for efficient computational approaches. We aim to learn parameterizations of the solutions that give more insight into the dynamics and/or increase computational efficiency. I will discuss our recent work using machine learning to develop models of the small scale behavior of spatio-temporal complex solutions, with the goal of maintaining accuracy albeit at a highly reduced computational cost relative to a full simulation. References: https://www.pnas.org/content/116/31/15344 and https://arxiv.org/pdf/2102.01010.pdf
