Speaker
Description
Partial differential equations (PDEs) arise to describe linear and nonlinear physical phenomena. In mechanical engineering, there is a particular need for new black box solvers of PDEs due to ever shorter product development times. The recent success in solving various PDEs with neural networks, particularly with physics-informed NNs (PINNs) suggests that they are a natural candidate for those solvers. Even though the range of PDEs that can be approximated seemingly well by PINNs is quite impressive, a rigorous a-posteriori error control is at least not straightforward. Classical PINNs are usually trained with loss functions based upon the pointwise residual. The advantage of this is, that the method is mesh-free, although it makes the error control difficult due to the fact that an error-residual relation is only available if the problem is well-posed. Therefore, the goal of our work is to certify PINN approximations for linear and nonlinear PDEs while preserving their broad applicability and keeping the additional cost of discretizing the underlying physical domain low. Given a trained PINN approximating the solution of a PDE, we embed the possibly complicated shaped domain into a simple shaped domain and construct an efficient computable upper bound for the error. The advantage of this ansatz is, that the discretization is straightforward. Moreover, to evaluate the error estimator we use Riesz representations and therefore the evaluation is efficient due to the simple shape of the domain. The method is applied to various elliptic and parabolic PDEs.
