# Convergence of physics-informed neural networks modeling time-harmonic wave fields

arXiv: https://arxiv.org/abs/2506.11395 · HTML: https://mlsanity.com/paper/2506.11395/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 18 May 2025.

## Builds on (most cited first, 11 of 11)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Learning Discrepancy Models From Experimental Data](https://mlsanity.com/paper/1909.08574/) (2019)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Loss Landscape Engineering via Data Regulation on PINNs](https://mlsanity.com/paper/2205.07843/) (2022)
- [Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance](https://mlsanity.com/paper/2303.03918/) (2023)
- [pyCFS-data: Data Processing Framework in Python for openCFS](https://mlsanity.com/paper/2405.03437/) (2024)
- [Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks](https://mlsanity.com/paper/2503.20222/) (2025)

## Abstract (arXiv)

Studying physics-informed neural networks (PINNs) for modeling partial differential equations to solve the acoustic wave field has produced promising results for simple geometries in two-dimensional domains. One option is to compute the time-harmonic wave field using the Helmholtz equation. Compared to existing numerical models, the physics-informed neural networks forward problem has to overcome several topics related to the convergence of the optimization toward the "true" solution. The topics reach from considering the physical dimensionality (from 2D to 3D), the modeling of realistic sources (from a self-similar source to a realistic confined point source), the modeling of sound-hard (Neumann) boundary conditions, and the modeling of the full wave field by considering the complex solution quantities. Within this contribution, we study 3D room acoustic cases at low frequency, varying the source definition and the number of boundary condition sets and using a complex speed of sound model to account for some degree of absorption. We assess the convergence behavior by looking at the loss landscape of the PINN architecture, the L2 error compared to a finite element reference simulation for each network architecture and configuration. The convergence studies showed that at least six training points per wavelength are necessary for accurate training and subsequent predictions of the PINN. The developments are part of an initiative aiming to model the low-frequency behavior of room acoustics, including absorbers.