# GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs

arXiv: https://arxiv.org/abs/2303.15849 · HTML: https://mlsanity.com/paper/2303.15849/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 28 Mar 2023.

## Builds on (most cited first, 14 of 14)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [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)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Not All Samples Are Created Equal: Deep Learning with Importance Sampling](https://mlsanity.com/paper/1803.00942/) (2018)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [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)
- [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)
- [Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulation](https://mlsanity.com/paper/2302.01529/) (2023)
- [Efficient Natural Gradient Descent Methods for Large-Scale PDE-Based Optimization Problems](https://mlsanity.com/paper/2202.06236/) (2022)

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

- [Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs](https://mlsanity.com/paper/2305.18702/) (2023)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths](https://mlsanity.com/paper/2501.15522/) (2025)
- [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025)
- [Consensus-based adaptive sampling and approximation for high-dimensional energy landscapes](https://mlsanity.com/paper/2311.05009/) (2023)
- [A Discrete Neural Operator with Adaptive Sampling for Surrogate Modeling of Parametric Transient Darcy Flows in Porous Media](https://mlsanity.com/paper/2512.03113/) (2025)

## Abstract (arXiv)

With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs can efficiently handle high-dimensional problems, but the accuracy is relatively low, especially for highly irregular problems. Inspired by the idea of adaptive finite element methods and incremental learning, we propose GAS, a Gaussian mixture distribution-based adaptive sampling method for PINNs. During the training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which are then trained together with historical data to speed up the convergence of the loss and achieve higher accuracy. Several numerical simulations on 2D and 10D problems show that GAS is a promising method that achieves state-of-the-art accuracy among deep solvers, while being comparable with traditional numerical solvers.