# Self-adaptive weighting and sampling for physics-informed neural networks

arXiv: https://arxiv.org/abs/2511.05452 · HTML: https://mlsanity.com/paper/2511.05452/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 35 references that are in the map.
- First posted on arXiv 7 Nov 2025.

## Builds on (most cited first, 20 of 20)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [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)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture](https://mlsanity.com/paper/2005.00615/) (2020)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)](https://mlsanity.com/paper/2209.09988/) (2022)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)
- [Ensemble learning for Physics Informed Neural Networks: a Gradient Boosting approach](https://mlsanity.com/paper/2302.13143/) (2023)
- [Physics-informed machine learning of redox flow battery based on a two-dimensional unit cell model](https://mlsanity.com/paper/2306.01010/) (2023)

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

- [ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms](https://mlsanity.com/paper/2512.03476/) (2025)
- [Synergizing Kolmogorov-Arnold Networks with Dynamic Adaptive Weighting for High-Frequency and Multi-Scale PDE Solutions](https://mlsanity.com/paper/2512.22283/) (2025)
- [Improving the accuracy of physics-informed neural networks via last-layer retraining](https://mlsanity.com/paper/2603.04672/) (2026)
- [Curvature-aware dynamic precision approach for physics-informed neural networks](https://mlsanity.com/paper/2606.04736/) (2026)
- [Multi-dimensional training-priority weighting based on physical information propagation paths: a unified residual-weighting framework for physics-informed neural networks](https://mlsanity.com/paper/2607.11094/) (2026)
- [Probabilistic Physics-Informed Neural Networks for Estimating Heterogeneous Elastic Properties from Low-Resolution and Noisy Displacement Data](https://mlsanity.com/paper/2607.14563/) (2026)

## Abstract (arXiv)

Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and efficiency. In this work, we introduce a hybrid adaptive sampling and weighting method to enhance the performance of physics-informed neural networks (PINNs). The adaptive sampling component identifies training points in regions where the solution exhibits rapid variation, while the adaptive weighting component balances the convergence rate across training points. Numerical experiments show that applying only adaptive sampling or only adaptive weighting is insufficient to consistently achieve accurate predictions, particularly when training points are scarce. Since each method emphasizes different aspects of the solution, their effectiveness is problem dependent. By combining both strategies, the proposed framework consistently improves prediction accuracy and training efficiency, offering a more robust approach for solving PDEs with PINNs.