mlsanityData to 14 Oct 2026

Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective

arXiv 2506.19805 ↗First posted 24 Jun 2025cs.LGcited by 6

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Abstract (arXiv)

Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by computational limitations, PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy. In this study, we proposed a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions. The empirical results show that our weighting scheme can reduce the relative L2 errors to a lower value.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 30 of its 52 references that are in the map. First posted on arXiv 24 Jun 2025.

Builds on

30 of 52 references.

References · 30

  1. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  2. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  4. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  5. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  6. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  7. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  8. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  9. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 137
  10. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  11. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  12. 2024Physics-Informed Diffusion Modelscited by 90
18 more
  1. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 87
  2. 2021Multi-Objective Loss Balancing for Physics-Informed Deep Learningcited by 87
  3. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 87
  4. 2023Residual-based attention in physics-informed neural networkscited by 72
  5. 2021Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equationscited by 68
  6. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 64
  7. 2022Failure-informed adaptive sampling for PINNscited by 64
  8. 2023PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networkscited by 64
  9. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  10. 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
  11. 2022Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equationscited by 11
  12. 2025Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Studycited by 11
  13. 2024Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)cited by 9
  14. 2022Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditionscited by 7
  15. 2025Complex Physics-Informed Neural Networkcited by 7
  16. 2024Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Modelscited by 4
  17. 2025An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networkscited by 4
  18. 2025Plug-and-Play Physics-informed Learning using Uncertainty Quantified Port-Hamiltonian Modelscited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 4; 2026: 2.

6 papers.

  1. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  2. 2026Lightweight Geometric Adaptation for Training Physics-Informed Neural Networkscited by 1
  3. 2025Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networkscited by 0
  4. 2025AutoBalance: An Automatic Balancing Framework for Training Physics-Informed Neural Networkscited by 0
  5. 2025RRaPINNs: Residual Risk-Aware Physics Informed Neural Networkscited by 0
  6. 2026From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNscited by 0