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

arXiv: https://arxiv.org/abs/2506.19805 · HTML: https://mlsanity.com/paper/2506.19805/

## 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 (most cited first, 30 of 30)

- [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)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [RoPINN: Region Optimized Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.14369/) (2024)
- [Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2206.09299/) (2022)
- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)](https://mlsanity.com/paper/2406.03172/) (2024)
- [Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions](https://mlsanity.com/paper/2208.08635/) (2022)
- [Complex Physics-Informed Neural Network](https://mlsanity.com/paper/2502.04917/) (2025)
- [Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Models](https://mlsanity.com/paper/2406.11809/) (2024)
- [An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks](https://mlsanity.com/paper/2501.11222/) (2025)
- [Plug-and-Play Physics-informed Learning using Uncertainty Quantified Port-Hamiltonian Models](https://mlsanity.com/paper/2504.17966/) (2025)

## Built on by (most cited first, 6 of 6)

- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [Lightweight Geometric Adaptation for Training Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.15392/) (2026)
- [Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks](https://mlsanity.com/paper/2508.19410/) (2025)
- [AutoBalance: An Automatic Balancing Framework for Training Physics-Informed Neural Networks](https://mlsanity.com/paper/2510.06684/) (2025)
- [RRaPINNs: Residual Risk-Aware Physics Informed Neural Networks](https://mlsanity.com/paper/2511.18515/) (2025)
- [From Non-Convex Self-Concordant Regularization to Scalable Quasi-Newton Training of PINNs](https://mlsanity.com/paper/2608.04206/) (2026)

## 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.