# Efficient Training of Physics-Informed Neural Networks with Direct Grid Refinement Algorithm

arXiv: https://arxiv.org/abs/2306.08293 · HTML: https://mlsanity.com/paper/2306.08293/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 13 references that are in the map.
- First posted on arXiv 14 Jun 2023.

## Builds on (most cited first, 10 of 10)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2205.01051/) (2022)
- [Dynamic weights enabled Physics-Informed Neural Network for simulating the mobility of Engineered Nano-particles in a contaminated aquifer](https://mlsanity.com/paper/2211.03525/) (2022)

## Abstract (arXiv)

This research presents the development of an innovative algorithm tailored for the adaptive sampling of residual points within the framework of Physics-Informed Neural Networks (PINNs). By addressing the limitations inherent in existing adaptive sampling techniques, our proposed methodology introduces a direct mesh refinement approach that effectively ensures both computational efficiency and adaptive point placement. Verification studies were conducted to evaluate the performance of our algorithm, showcasing reasonable agreement between the model based on our novel approach and benchmark model results. Comparative analyses with established adaptive resampling techniques demonstrated the superior performance of our approach, particularly when implemented with higher refinement factor. Overall, our findings highlight the enhancement of simulation accuracy achievable through the application of our adaptive sampling algorithm for Physics-Informed Neural Networks.