Convergence and error control of consistent PINNs for elliptic PDEs

We provide an a priori analysis of a certain class of numerical methods, commonly referred to as collocation methods, for solving elliptic boundary value problems. They begin with information in the form of point values of the right side f of such equations and point values of the boundary function g and utilize only this information to numerically approximate the solution u of the Partial Differential Equation (PDE). For such a method to provide an approximation to u with guaranteed error bounds, additional assumptions on f and g, called model class assumptions, are needed. We determine the best error (in the energy norm) of approximating u, in terms of the number of point samples m, under all Besov class model assumptions for the right hand side $f$ and boundary g. We then turn to the study of numerical procedures and asks whether a proposed numerical procedure (nearly) achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to $u$ by minimizing a specified data driven loss function over a set $\Sigma$ which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a correct data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a loss function $L^*$ is proposed. We also address the recent methods of Physics Informed Neural Networks (PINNs). Minimization of the new loss $L^*$ over neural network spaces $\Sigma$ is referred to as consistent PINNs (CPINNs). We prove that CPINNs provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $\Sigma$ has sufficient approximation capabilities. Finally, numerical examples illustrating the benefits of the CPINNs are given.