Physics-Informed Neural Networks for Laplace and Steklov Eigenvalue Problems

Date

2026-9-03

Author

Zengin, Kerem

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This thesis investigates Physics-Informed Neural Network (PINN) approximations of the Laplace and Steklov eigenvalue problems (EVPs). For the Laplace EVP, the principal formulation choices, which appear in various forms in the existing literature, are brought together in a systematic comparative analysis. These choices include soft and hard boundary enforcement, explicit and Rayleigh-induced eigenvalue representations, and simultaneous and sequential strategies for computing multiple eigenmodes. Their effects are evaluated under a common computational protocol using boundary admissibility, spectral accuracy, Gram-matrix orthogonality, and optimization stability as the main assessment criteria. The comparison identifies a reference formulation based on hard boundary enforcement, Rayleigh-induced eigenvalues, and sequential computation with orthogonality control. Applied to two- and three-dimensional benchmark domains, this formulation approximates the target eigenpairs accurately while maintaining mutual orthogonality among the computed modes. Building on the formulation principles established for the Laplace EVP, the study is then extended to the Steklov EVP, in which the spectral parameter enters through a boundary condition rather than the interior differential equation. The Steklov spectral parameter multiplies the unknown boundary trace; hard boundary enforcement therefore cannot encode the spectral relation in the neural trial function, and a residual-based formulation with soft enforcement of the boundary condition is required. Consequently, a residual-based physics-informed approach is designed to approximate Steklov eigenpairs with the computed modes normalized and mutually orthogonalized on the Steklov boundary. Numerical experiments on benchmark domains show accurate approximation of the Steklov eigenvalues and eigenfunctions, together with preservation of this boundary orthogonality. To the best of our knowledge, such a residual-based physics-informed neural approximation of the Steklov EVP is among the first in this direction. Overall, the results establish a common physics-informed framework for the numerical approximation of both Laplace and Steklov EVPs.

Subject Keywords

Physics-Informed Neural Networks, Laplace EVP, Steklov EVP

URI

https://hdl.handle.net/11511/120648

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Graduate School of Applied Mathematics, Thesis