## Some Thoughts on Physics Informed Neural Networks

### Wolfgang Dahmen

South Carolina University

Employing Deep Learning concepts to "learn" physical laws, has been recently attracting significant attention. In particular, so called "Physics Informed Neural Networks" (PINN) refers to a paradigm where the training of model surrogates is based on empirical risks that require only point-wise evaluation of residuals. This avoids the expensive computation of a sufficiently large number of training data, typically given in terms of high fidelity approximations of model states.

The core issue addressed in this talk is the prediction capability of such methods for models given in terms of parameter-dependent families of partial differential equations. Related specific questions concern, for instance, the choice of "variationally correct" training risks, that convey certifiable information about the achieved accuracy in problem relevant metrics, the role of a priori versus a posteriori error bounds, connections with Generative Adversarial Networks, as well as related implications on training strategies and network adaptation.