A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

TL;DR

This paper introduces a machine learning-based decision-making approach to dynamically adapt Hermite spectral methods for solving time-dependent partial differential equations on unbounded domains, improving accuracy especially in under-resolved scenarios.

Contribution

It proposes a novel automatic decision-making process using deep neural networks and support vector machines to optimize Hermite spectral approximations in PDEs.

Findings

Successful numerical tests on a 1D problem

Effective dynamic adaptation of approximation space

Potential for application to complex PDE problems

Abstract

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.