Uncertainty Quantification for Parameter dependent Partial Differential Equations using Deep Neural Networks

In many fields, such as engineering, healthcare and finance, Uncertainty Quantification (UQ) plays an essential role in the development of robust and reliable models. UQ techniques often involve many-query algorithms, such as those typical of Monte Carlo methods, Frequentist and Bayesian inference. As a consequence, in the case of highdimensional and complex systems such as those driven by PDE models, UQ becomes extremely challenging and computationally expensive. One possible solution is to reduce the computational cost by introducing a suitable surrogate model, which approximates the underlying Full Order Model at a negligible compromise in accuracy. Here, we consider models described by parameter dependent PDEs and we focus on the use of Deep Neural Networks (DNNs) as model surrogates. In order to fully exploit the nonlinear capabilities of DNNs, we base the construction of the network on the concept of manifold-width. After discussing the theory and the driving ideas of our approach, such as minimal AutoEncoders and mesh-informed layers, we present some numerical examples with applications to UQ and microcirculation of oxygen.