Data-Driven Model Reduction, Scientific Frontiers, and Applications ()
- Texas A&M University
- College Station, TX
- Joe C. Richardson Petroleum Engineering Building (RICH) 910
Jonathan Siegel, Department of Mathematics
Convergence and Error Control of Consistent PINNs for Elliptic PDEs
Abstract
We study the convergence rate, in terms of the number of collocation points, of Physics-Informed Neural Networks (PINNs) for the solution of elliptic PDEs. Specifically, given Sobolev space assumptions on the right-hand side of the PDE and on the boundary values, we determine the minimal number of collocation points required to achieve a given accuracy. These results apply more generally to any collocation method which only makes use of point values. Based upon this theory we introduce novel PINNs loss functions which we call consistent PINNs. We derive an a posteriori error estimator based upon the consistent PINNs loss functions. Finally, we present numerical experiments which demonstrate that consistent PINNs result in significantly improved error compared with the original PINNs loss function.