  The accuracy and effectiveness of Hermite spectral methods for the numerical
discretization of partial differential equations on unbounded domains, are
strongly affected by the amplitude of the Gaussian weight function employed to
describe the approximation space. This is particularly true if the problem is
under-resolved, i.e., there are no enough degrees of freedom. The issue becomes
even more crucial when the equation under study is time-dependent, forcing in
this way the choice of Hermite functions where the corresponding weight depends
on time. In order to adapt dynamically the approximation space, it is here
proposed an automatic decision-making process that relies on machine learning
techniques, such as deep neural networks and support vector machines. The
algorithm is numerically tested with success on a simple 1D problem, but the
main goal is its exportability in the context of more serious applications.
