Machine learning and partial differential equations (PDEs)
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The tools from machine learning (ML) offer exciting new prospects in scientific computing in general, and the numerical solution of PDEs in particular. In this talk, I will discuss some recent results in this context, by focusing on problems with a variational formulation, and discussing three main ingredients of their solution by ML: (i) approximation quality, i.e. how accurate the representation of the solution by a neural network can in principle be; (ii) optimization, i.e. how effective are the methods to train the parameter of the network; and (iii) generalization error, i.e. how much data is necessary to obtain a solution accurate also outside this data set. In particular I will show that this third aspect typically requires using importance sampling methods for data acquisition. These results will be illustrated on eigenvalue problems in high dimension. This is joint work with Grant Rotskoff. Zoom: https://arizona.zoom.us/j/91826900125 Password: "Locute"