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

arXiv 2205.06704 ↗First posted 13 May 2022math.NA cs.AI cs.NAcited by 18

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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.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 12 of its 34 references that are in the map. First posted on arXiv 13 May 2022.

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12 of 34 references.

References · 12

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  3. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  4. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 360
  5. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  6. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  8. 2020Hyper-Parameter Optimization: A Review of Algorithms and Applicationscited by 114
  9. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  10. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61
  11. 2017A Bridge Between Hyperparameter Optimization and Learning-to-learncited by 6
  12. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 2; 2024: 8; 2025: 3; 2026: 3.

18 papers.

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  1. 2022Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Mediacited by 8
  2. 2024Binary structured physics-informed neural networks for solving equations with rapidly changing solutionscited by 8
  3. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  4. 2023Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performancecited by 5
  5. 2023h-analysis and data-parallel physics-informed neural networkscited by 4
  6. 2024Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computationcited by 4
  7. 2022Physics-informed neural networks for operator equations with stochastic datacited by 1
  8. 2024Estimate Epidemiological Parameters given Partial Observations based on Algebraically Observable PINNscited by 1
  9. 2024Surface Flux Transport Modeling using Physics Informed Neural Networkscited by 1
  10. 2025BO-SA-PINNs: Self-adaptive physics-informed neural networks based on Bayesian optimization for automatically designing PDE solverscited by 1
  11. 2024Taper-based scattering formulation of the Helmholtz equation to improve the training process of Physics-Informed Neural Networkscited by 0
  12. 2024Revising the Structure of Recurrent Neural Networks to Eliminate Numerical Derivatives in Forming Physics Informed Loss Terms with Respect to Timecited by 0
6 more
  1. 2024A Physics Informed Neural Network (PINN) Methodology for Coupled Moving Boundary PDEscited by 0
  2. 2025Algebraically Observable Physics-Informed Neural Network and its Application to Epidemiological Modellingcited by 0
  3. 2025Benchmarking Physics-Informed Neural Networks and Boundary Elements Methods for Wave Scatteringcited by 0
  4. 2026Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennescited by 0
  5. 2026Virtual boundary integral neural network for three-dimensional exterior acoustic problemscited by 0
  6. 2026Evolutionary Two-Stage Hyperparameter Optimization Strategies for Physics-Informed Neural Networkscited by 0