# Adaptive importance sampling for Deep Ritz

arXiv: https://arxiv.org/abs/2310.17185 · HTML: https://mlsanity.com/paper/2310.17185/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 19 references that are in the map.
- First posted on arXiv 26 Oct 2023.

## Builds on (most cited first, 11 of 11)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [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)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [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)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)

## Abstract (arXiv)

We introduce an adaptive sampling method for the Deep Ritz method aimed at solving partial differential equations (PDEs). Two deep neural networks are used. One network is employed to approximate the solution of PDEs, while the other one is a deep generative model used to generate new collocation points to refine the training set. The adaptive sampling procedure consists of two main steps. The first step is solving the PDEs using the Deep Ritz method by minimizing an associated variational loss discretized by the collocation points in the training set. The second step involves generating a new training set, which is then used in subsequent computations to further improve the accuracy of the current approximate solution. We treat the integrand in the variational loss as an unnormalized probability density function (PDF) and approximate it using a deep generative model called bounded KRnet. The new samples and their associated PDF values are obtained from the bounded KRnet. With these new samples and their associated PDF values, the variational loss can be approximated more accurately by importance sampling. Compared to the original Deep Ritz method, the proposed adaptive method improves accuracy, especially for problems characterized by low regularity and high dimensionality. We demonstrate the effectiveness of our new method through a series of numerical experiments.