  In recent years, a plethora of methods combining deep neural networks and
partial differential equations have been developed. A widely known and popular
example are physics-informed neural networks. They solve forward and inverse
problems involving partial differential equations in terms of a neural network
training problem. We apply physics-informed neural networks as well as the
finite element method to estimate the diffusion coefficient governing the long
term, i.e. over days, spread of molecules in the human brain from a novel
magnetic resonance imaging technique. Synthetic testcases are created to
demonstrate that the standard formulation of the physics-informed neural
network faces challenges with noisy measurements in our application. Our
numerical results demonstrate that the residual of the partial differential
equation after training needs to be small in order to obtain accurate recovery
of the diffusion coefficient. To achieve this, we apply several strategies such
as tuning the weights and the norms used in the loss function as well as
residual based adaptive refinement and exchange of residual training points. We
find that the diffusion coefficient estimated with PINNs from magnetic
resonance images becomes consistent with results from a finite element based
approach when the residuum after training becomes small. The observations
presented in this work are an important first step towards solving inverse
problems on observations from large cohorts of patients in a semi-automated
fashion with physics-informed neural networks.
