# Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks

arXiv: https://arxiv.org/abs/2503.20222 · HTML: https://mlsanity.com/paper/2503.20222/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 71 references that are in the map.
- First posted on arXiv 26 Mar 2025.

## Builds on (most cited first, 16 of 16)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [A Survey of Deep Learning Applications to Autonomous Vehicle Control](https://mlsanity.com/paper/1912.10773/) (2019)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Physics-informed neural networks for solving forward and inverse problems in complex beam systems](https://mlsanity.com/paper/2303.01055/) (2023)
- [Improving physics-informed neural networks with meta-learned optimization](https://mlsanity.com/paper/2303.07127/) (2023)
- [A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate](https://mlsanity.com/paper/2102.02617/) (2021)
- [A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound](https://mlsanity.com/paper/2010.15761/) (2020)
- [Augmenting a Physics-Informed Neural Network for the 2D Burgers Equation by Addition of Solution Data Points](https://mlsanity.com/paper/2301.07824/) (2023)
- [A study on native American English speech recognition by Indian listeners with varying word familiarity level](https://mlsanity.com/paper/2112.04151/) (2021)
- [Neural network based approach for solving problems in plane wave duct acoustics](https://mlsanity.com/paper/2405.04603/) (2024)
- [Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media](https://mlsanity.com/paper/2305.06802/) (2023)

## Built on by (most cited first, 1 of 1)

- [Convergence of physics-informed neural networks modeling time-harmonic wave fields](https://mlsanity.com/paper/2506.11395/) (2025)

## Abstract (arXiv)

Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing-gradient problem that occurs when solving the Helmholtz equation. In this paper, a formulation is presented that addresses this difficulty. The problem of solving the two-dimensional Helmholtz equation with the prescribed boundary conditions is posed as an unconstrained optimization problem using trial solution method. According to this method, a trial neural network that satisfies the given boundary conditions prior to the training process is constructed using the technique of transfinite interpolation and the theory of R-functions. This ansatz is initially applied to the rectangular domain and later extended to the circular and elliptical domains. The acoustic field predicted from the proposed formulation is compared with that obtained from the two-dimensional finite element methods. Good agreement is observed in all three domains considered. Minor limitations associated with the proposed formulation and their remedies are also discussed.