Abstract
We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we compute localized bounded states at the lower end of the spectrum. We present several deep network architectures that predict the localization of bounded states from a sample of a potential. For tackling higher dimensional problems, we consider a class of physics-informed deep dense networks. In particular, we focus on the interpretability of the proposed approaches. Deep network is used as a general reduced order model that describes the nonlinear connection between the potential and the ground state. The performance of the surrogate reduced model is controlled by an error estimator and the model is updated if necessary. Finally, we present a host of experiments to measure the accuracy and performance of the proposed algorithm.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional clai-
To benchmark the accuracy of the VPINN approximations we have solved the problem to high relative accuracy using the Chebyshev spectral method as implemented in the package chebfun [43,44]
We have presented two types of neural network architectures
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional clai-
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