  We propose a nonlinear reduced basis method for the efficient approximation
of parametrized partial differential equations (PDEs), exploiting kernel proper
orthogonal decomposition (KPOD) for the generation of a reduced-order space and
neural networks for the evaluation of the reduced-order approximation. In
particular, we use KPOD in place of the more classical POD, on a set of
high-fidelity solutions of the problem at hand to extract a reduced basis. This
method provides a more accurate approximation of the snapshots' set featuring a
lower dimension, while maintaining the same efficiency as POD. A neural network
(NN) is then used to find the coefficients of the reduced basis by following a
supervised learning paradigm and shown to be effective in learning the map
between the time/parameter values and the projection of the high-fidelity
snapshots onto the reduced space. In this NN, both the number of hidden layers
and the number of neurons vary according to the intrinsic dimension of the
differential problem at hand and the size of the reduced space. This adaptively
built NN attains good performances in both the learning and the testing phases.
Our approach is then tested on two benchmark problems, a one-dimensional wave
equation and a two-dimensional nonlinear lid-driven cavity problem. We finally
compare the proposed KPOD-NN technique with a POD-NN strategy, showing that
KPOD allows a reduction of the number of modes that must be retained to reach a
given accuracy in the reduced basis approximation. For this reason, the NN
built to find the coefficients of the KPOD expansion is smaller, easier and
less computationally demanding to train than the one used in the POD-NN
strategy.
