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A PINNs approach for the computation of eigenvalues in elliptic problems

Julian Fernandez Bonder, Ariel M. Salort

TL;DR

A key feature of this approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without requiring the prior computation of lower ones.

Abstract

In this paper, we propose a method for computing eigenvalues of elliptic problems using Deep Learning techniques. A key feature of our approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without requiring the prior computation of lower ones. Moreover, the method can be easily adapted to handle nonlinear eigenvalue problems.

A PINNs approach for the computation of eigenvalues in elliptic problems

TL;DR

A key feature of this approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without requiring the prior computation of lower ones.

Abstract

In this paper, we propose a method for computing eigenvalues of elliptic problems using Deep Learning techniques. A key feature of our approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without requiring the prior computation of lower ones. Moreover, the method can be easily adapted to handle nonlinear eigenvalue problems.

Paper Structure

This paper contains 13 sections, 1 theorem, 30 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Our approach
  3. Implementation details
  4. Further examples
  5. Higher eigenvalues
  6. Higher dimensions
  7. Different geometries
  8. Different potentials
  9. Nonlinear problems
  10. Code availability
  11. Conclusions
  12. acknowledgments
  13. The minimization problem

Key Result

Theorem 1

Let $V(x)$ be as in conf.V with $\Omega\subset \mathbb{R}^n$ bounded. Given $E\ge 0$, there exists $u_E\in H^2(\Omega)\cap H^1_0(\Omega)$ such that where $\Lambda(u, E)$ is given in Lambda.

Figures (15)

  • Figure 1: A fully connected feedforward NN with an $n$-dimensional input and a 1-dimensional output.
  • Figure 2: The loss curve $\Lambda(E)$ in $2-$dimensional space with a confined potential in the unit disk.
  • Figure 3: The loss curve $\bar{\Lambda}(E)$ in two-dimensional space with a confined potential in the unit disk compared with the theoretical upper bound.
  • Figure 4: The first four computed eigenfunctions of \ref{['eigen2']} in the $2-$dimensional unit disk.
  • Figure 5: The loss curve $\bar{\Lambda}(E)$ in two-dimensional space in the unit square computed in the interval [44,55].
  • ...and 10 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof