Regularity and Tailored Regularization of Deep Neural Networks, with application to parametric PDEs in uncertainty quantification

In this paper we consider Deep Neural Networks (DNNs) with a smooth activation function as surrogates for high-dimensional functions that are somewhat smooth but costly to evaluate. We consider the standard (non-periodic) DNNs as well as propose a new model of periodic DNNs which are especially suited for a class of periodic target functions when Quasi-Monte Carlo lattice points are used as training points. We study the regularity of DNNs by obtaining explicit bounds on all mixed derivatives with respect to the input parameters. The bounds depend on the neural network parameters as well as the choice of activation function. By imposing restrictions on the network parameters to match the regularity features of the target functions, we prove that DNNs with $N$ tailor-constructed lattice training points can achieve the generalization error (or $L_2$ approximation error) bound ${\tt tol} + \mathcal{O}(N^{-r/2})$, where ${\tt tol}\in (0,1)$ is the tolerance achieved by the training error in practice, and $r = 1/p^*$, with $p^*$ being the ``summability exponent'' of a sequence that characterises the decay of the input variables in the target functions, and with the implied constant independent of the dimensionality of the input data. We apply our analysis to popular models of parametric elliptic PDEs in uncertainty quantification. In our numerical experiments, we restrict the network parameters during training by adding tailored regularization terms, and we show that for an algebraic equation mimicking the parametric PDE problems the DNNs trained with tailored regularization perform significantly better.