# Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems

arXiv: https://arxiv.org/abs/2205.06704 · HTML: https://mlsanity.com/paper/2205.06704/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 34 references that are in the map.
- First posted on arXiv 13 May 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Hyper-Parameter Optimization: A Review of Algorithms and Applications](https://mlsanity.com/paper/2003.05689/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [A Bridge Between Hyperparameter Optimization and Learning-to-learn](https://mlsanity.com/paper/1712.06283/) (2017)
- [Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantification](https://mlsanity.com/paper/2101.07023/) (2021)

## Built on by (most cited first, 19 of 19)

- [Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media](https://mlsanity.com/paper/2212.08965/) (2022)
- [Binary structured physics-informed neural networks for solving equations with rapidly changing solutions](https://mlsanity.com/paper/2401.12806/) (2024)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance](https://mlsanity.com/paper/2303.03918/) (2023)
- [Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computation](https://mlsanity.com/paper/2405.11208/) (2024)
- [h-analysis and data-parallel physics-informed neural networks](https://mlsanity.com/paper/2302.08835/) (2023)
- [Physics-informed neural networks for operator equations with stochastic data](https://mlsanity.com/paper/2211.10344/) (2022)
- [Estimate Epidemiological Parameters given Partial Observations based on Algebraically Observable PINNs](https://mlsanity.com/paper/2407.12598/) (2024)
- [Surface Flux Transport Modeling using Physics Informed Neural Networks](https://mlsanity.com/paper/2409.01744/) (2024)
- [BO-SA-PINNs: Self-adaptive physics-informed neural networks based on Bayesian optimization for automatically designing PDE solvers](https://mlsanity.com/paper/2504.09804/) (2025)
- [Taper-based scattering formulation of the Helmholtz equation to improve the training process of Physics-Informed Neural Networks](https://mlsanity.com/paper/2404.09794/) (2024)
- [Revising the Structure of Recurrent Neural Networks to Eliminate Numerical Derivatives in Forming Physics Informed Loss Terms with Respect to Time](https://mlsanity.com/paper/2409.10388/) (2024)
- [A Physics Informed Neural Network (PINN) Methodology for Coupled Moving Boundary PDEs](https://mlsanity.com/paper/2409.10910/) (2024)
- [Algebraically Observable Physics-Informed Neural Network and its Application to Epidemiological Modelling](https://mlsanity.com/paper/2508.04590/) (2025)
- [Benchmarking Physics-Informed Neural Networks and Boundary Elements Methods for Wave Scattering](https://mlsanity.com/paper/2509.12483/) (2025)
- [Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes](https://mlsanity.com/paper/2603.27936/) (2026)
- [Virtual boundary integral neural network for three-dimensional exterior acoustic problems](https://mlsanity.com/paper/2604.18636/) (2026)
- [Evolutionary Two-Stage Hyperparameter Optimization Strategies for Physics-Informed Neural Networks](https://mlsanity.com/paper/2606.20442/) (2026)
- [Hyperparameter selection for equation learning with biologically-informed neural networks](https://mlsanity.com/paper/2610.02954/) (2026)

## Abstract (arXiv)

We consider physics-informed neural networks (PINNs) [Raissi et al., J.~Comput. Phys. 278 (2019) 686-707] for forward physical problems. In order to find optimal PINNs configuration, we introduce a hyper-parameter optimization (HPO) procedure via Gaussian processes-based Bayesian optimization. We apply the HPO to Helmholtz equation for bounded domains and conduct a thorough study, focusing on: (i) performance, (ii) the collocation points density r and (iii) the frequency kappa, confirming the applicability and necessity of the method. Numerical experiments are performed in two and three dimensions, including comparison to finite element methods.