Authors: Geetanjli and Kirankumar R Hiremath

Journal:  IoP Journal of Machine Learning: Engineering 

Link: https://doi.org/10.1088/3049-4761/ae510e

DOI:  https://doi.org/10.1088/3049-4761/ae510e

Year: 2026

Publisher:  Institute of Physics

Abstract: Physics-informed neural networks (PINNs) have emerged as powerful deep-learning frameworks for solving partial differential equations by directly embedding physical laws into the learning process. Leveraging this, we propose a PINN-based approach to solve the Helmholtz eigenvalue problem that arises in the analysis of waveguide (WG) modes. Specifically, we study eigenmodes in WG geometries that are difficult to handle using analytical and numerical methods such as the finite-difference method. We designed a flexible PINN architecture that learns both the mode profile and propagation constant, guided solely by the Helmholtz equation and Dirichlet boundary conditions. To accelerate convergence and maintain training stability, we adopted a carefully chosen parameter setup, including the informed initialization of the propagation constant, suitable activation functions, and optimized learning rates, which enabled the model to converge effectively without relying on labeled data or analytical solutions. Our results demonstrate the accurate recovery of mode profiles and eigenvalues for different geometries and boundary conditions. These initial results highlight the potential of PINNs for optical mode analysis and motivate further development of complex photonic structures.