# Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulation

arXiv: https://arxiv.org/abs/2302.01529 · HTML: https://mlsanity.com/paper/2302.01529/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 21 references that are in the map.
- First posted on arXiv 3 Feb 2023.

## Builds on (most cited first, 13 of 13)

- [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)
- [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)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2205.01051/) (2022)

## Built on by (most cited first, 13 of 13)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms](https://mlsanity.com/paper/2512.03476/) (2025)
- [GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs](https://mlsanity.com/paper/2303.15849/) (2023)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation](https://mlsanity.com/paper/2405.19256/) (2024)
- [An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks](https://mlsanity.com/paper/2501.11222/) (2025)
- [Learn Singularly Perturbed Solutions via Homotopy Dynamics](https://mlsanity.com/paper/2502.00488/) (2025)
- [Adaptive importance sampling for Deep Ritz](https://mlsanity.com/paper/2310.17185/) (2023)
- [Two-scale Neural Networks for Partial Differential Equations with Small Parameters](https://mlsanity.com/paper/2402.17232/) (2024)
- [Auto-Adaptive PINNs with Applications to Phase Transitions](https://mlsanity.com/paper/2510.23999/) (2025)
- [A hybrid analytical-PINN model for subsurface simulation of geothermal heat exchangers in heterogeneous underground](https://mlsanity.com/paper/2607.12271/) (2026)

## Abstract (arXiv)

This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks (FI-PINNs). In our previous work citegao2022failure, we have presented an adaptive sampling framework by using the failure probability as the posterior error indicator, where the truncated Gaussian model has been adopted for estimating the indicator. In this work, we present two novel extensions to FI-PINNs. The first extension consist in combining with a re-sampling technique, so that the new algorithm can maintain a constant training size. This is achieved through a cosine-annealing, which gradually transforms the sampling of collocation points from uniform to adaptive via training progress. The second extension is to present the subset simulation algorithm as the posterior model (instead of the truncated Gaussian model) for estimating the error indicator, which can more effectively estimate the failure probability and generate new effective training points in the failure region. We investigate the performance of the new approach using several challenging problems, and numerical experiments demonstrate a significant improvement over the original algorithm.