BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:r-adaptivity\, deep learning and optimal transport - Christopher B udd (University of Bath) DTSTART:20211208T170000Z DTEND:20211208T183000Z UID:TALK165148@talks.cam.ac.uk DESCRIPTION:PINNS (physics inspired neural networks) have recently become popular as a means of solving ODEs and PDES by using the tools of deep lea rning. They have both shown promise for solving some differential equation s\, and have struggled to solve others. Whilst advertised as being 'mesh f ree methods' they do rely on the use of collocation points. The accuracy o f the numerical solution of PDEs using Finite Element methods depends cruc ially on the choice of an appropriate mesh. This can be obtained an r-adap tive strategy\, which equidistributes the error over the mesh elements bas ed on a-priori/posteriori knowledge of the solution. The core of this talk will describe how r-adaptivity can be useful in the context of Deep Learn ing. First\, we will show that a one-dimensional mesh can be equidistribut ed by training a feed forward Neural Network. This approach yields better results than other standard numerical methods. We will then explain the tr aining process of Physics-informed Neural Networks (PINNs) for solving Bou ndary value problems (BVPs) and show numerical results for a reaction-diff usion and convection-dominated equation. It appears that PINNs fail to be trained in the latter case unless the homotropy method is employed. Finall y\, we will introduce the Deep-Ritz-Network (DRN) for solving the Poisson equation on a non-convex 2-dimensional domain. If the collocation points a re uniformly random sampled and fixed for the entire training process\, we obtain a solution with poor accuracy. On the contrary\, the adoption of a n Optimal Transport \; strategy\, which determines the 'optimal' collo cation points\, results in a more stable training process and a much more accurate solution LOCATION:Seminar Room 2\, Newton Institute END:VEVENT END:VCALENDAR