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We propose a nonlinear reduced basis method for the efficient approximationof parametrized partial differential equations (PDEs), exploiting kernel properorthogonal decomposition (KPOD) for the generation of a reduced-order space andneural networks for the evaluation of the reduced-order approximation. Inparticular, we use KPOD in place of the more classical POD, on a set ofhigh-fidelity solutions of the problem at hand to extract a reduced basis. Thismethod provides a more accurate approximation of the snapshots' set featuring alower 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 asupervised learning paradigm and shown to be effective in learning the mapbetween the time/parameter values and the projection of the high-fidelitysnapshots onto the reduced space. In this NN, both the number of hidden layersand the number of neurons vary according to the intrinsic dimension of thedifferential problem at hand and the size of the reduced space. This adaptivelybuilt NN attains good performances in both the learning and the testing phases.Our approach is then tested on two benchmark problems, a one-dimensional waveequation and a two-dimensional nonlinear lid-driven cavity problem. We finallycompare the proposed KPOD-NN technique with a POD-NN strategy, showing thatKPOD allows a reduction of the number of modes that must be retained to reach agiven accuracy in the reduced basis approximation. For this reason, the NNbuilt to find the coefficients of the KPOD expansion is smaller, easier andless computationally demanding to train than the one used in the POD-NNstrategy.

Non intrusive reduced order modeling of parametrized PDEs by kernel POD and neural networks