Solution discovery in fluid dynamics using neural networks

I will discuss two examples of utilizing neural networks (NN) to find solutions to partial differential equations (PDEs). The first concerns the search for self-similar blow-up solutions of the Euler equations (Wang-Lai-Gómez-Serrano-Buckmaster, Phys. Rev. Lett. 2023). The second application uses PDE-constrained NNs as an inverse method in geophysics. Whether an inviscid incompressible fluid, described by the 3-dimensional Euler equations, can develop singularities in finite time is an open question in mathematical fluid dynamics. We employ NNs to find a numerical self-similar blow-up solution for the incompressible 3-dimensional Euler equations with a cylindrical boundary. In the second part of the talk, I will discuss how PDE-constrained NNs trained with real world data from Antarctica can help discover the fluid rhoelogy that govern ice-shelf dynamics. Despite its importance in governing the flow of glaciers into the ocean, the rheology of glacial ice, a crucial material property, cannot be directly measured in the field. Several geophysical-scale phenomena could potentially cause the laboratory-derived rheology of ice to deviate from itss behavior in the field. Using NN with data measured from space, here we infer glacial rheology that differ from those commonly assumed in climate simulations. This demonstrates the need to reassess the rheology of geophysical complex fluids beyond the laboratory setting.