Deep spectral computations in linear and nonlinear diffusion problems

TL;DR

This paper introduces a neural network-based variational framework for solving complex eigenvalue problems in high-dimensional diffusion operators, outperforming classical methods especially in challenging non-self adjoint and nonlinear cases.

Contribution

It presents a novel neural network approach capable of handling non-self adjoint, deep spectrum, multiple eigenmodes, and nonlinear eigenvalue problems in high dimensions.

Findings

Successfully computed eigenpairs for 10D Kolmogorov operator

Accurately approximated 5D Schrödinger operator with 32 metastable states

Solved nonlinear Gelfand superlinear problem in 4D with complex domains

Abstract

We propose a flexible machine-learning framework for solving eigenvalue problems of diffusion operators in moderately large dimension. We improve on existing Neural Networks (NNs) eigensolvers by demonstrating our approach ability to compute (i) eigensolutions for non-self adjoint operators with small diffusion (ii) eigenpairs located deep within the spectrum (iii) computing several eigenmodes at once (iv) handling nonlinear eigenvalue problems. To do so, we adopt a variational approach consisting of minimizing a natural cost functional involving Rayleigh quotients, by means of simple adiabatic technics and multivalued feedforward neural parametrisation of the solutions. Compelling successes are reported for a 10-dimensional eigenvalue problem corresponding to a Kolmogorov operator associated with a mixing Stepanov flow. We moreover show that the approach allows for providing accurate eigensolutions for a 5-D Schr\"odinger operator having $32$ metastable states. In addition, we address the so-called Gelfand superlinear problem having exponential nonlinearities, in dimension $4$, and for nontrivial domains exhibiting cavities. In particular, we obtain NN-approximations of high-energy solutions approaching singular ones. We stress that each of these results are obtained using small-size neural networks in situations where classical methods are hopeless due to the curse of dimensionality. This work brings new perspectives for the study of Ruelle-Pollicot resonances, dimension reduction, nonlinear eigenvalue problems, and the study of metastability when the dynamics has no potential.