Active Learning Based Sampling for High-Dimensional Nonlinear Partial Differential Equations

Gao, Weizhen; Wang, Chunmei

doi:10.48550/arxiv.2112.13988

Cited by 2 publications

(4 citation statements)

References 41 publications

(54 reference statements)

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“…During the preparation of this paper, a few new studies appeared [38,39,40,41,42,43,44] that also proposed modified versions of RAR or PDF-based resampling. Most of these methods are special cases of the proposed RAD and RAR-D, and our methods can achieve better performance.…”

Section: Related Work and Our Contributionsmentioning

confidence: 99%

“…During the preparation of this paper, a few new papers appeared [38,39,40,41,42,43,44] that also proposed similar methods. Here, we summarize the similarities and differences between these studies.…”

Section: Comparison With Related Workmentioning

confidence: 99%

“…• The method proposed by Gao et al [40] (in December 2021) is a special case of RAD by choosing c = 0.…”

Section: Comparison With Related Workmentioning

confidence: 99%

See 2 more Smart Citations

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Wu¹,

Zhu²,

Tan³

et al. 2022

Preprint

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Physics-informed neural networks (PINNs) have shown to be an effective tool for solving both forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network using automatic differentiation, and this PDE loss is evaluated at a set of scattered spatio-temporal points (called residual points). The location and distribution of these residual points are highly important to the performance of PINNs. However, in the existing studies on PINNs, only a few simple residual point sampling methods have mainly been used. Here, we present a comprehensive study of two categories of sampling for PINNs: non-adaptive uniform sampling and adaptive nonuniform sampling. We consider six uniform sampling methods, including (1) equispaced uniform grid, (2) uniformly random sampling, (3) Latin hypercube sampling, (4) Halton sequence, (5) Hammersley sequence, and (6) Sobol sequence. We also consider a resampling strategy for uniform sampling. To improve the sampling efficiency and the accuracy of PINNs, we propose two new residual-based adaptive sampling methods: residual-based adaptive distribution (RAD) and residual-based adaptive refinement with distribution (RAR-D), which dynamically improve the distribution of residual points based on the PDE residuals during training. Hence, we have considered a total of 10 different sampling methods, including six non-adaptive uniform sampling, uniform sampling with resampling, two proposed adaptive sampling, and an existing adaptive sampling. We extensively tested the performance of these sampling methods for four forward problems and two inverse problems in many setups. Our numerical results presented in this study are summarized from more than 6000 simulations of PINNs. We show that the proposed adaptive sampling methods of RAD and RAR-D significantly improve the accuracy of PINNs with fewer residual points for both forward and inverse problems. The results obtained in this study can also be used as a practical guideline in choosing sampling methods.

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Section: Related Work and Our Contributionsmentioning

confidence: 99%

Section: Comparison With Related Workmentioning

confidence: 99%

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A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Wu¹,

Zhu²,

Tan³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…But when the dimension is high or the small-scale feature regions of solution are small, it is difficult for the candidate points to fall into the regions with large residuals, which affects the efficiency of the algorithm. In [6], the idea of using p-th power of the absolute residuals as the unnormalized probability distribution has been proposd. Using Metropolis-Hastings method or self-normalized sampling method to sample collocation points is another highlight of this paper.…”

Section: Introductionmentioning

confidence: 99%

MCMC-PINNs: A modified Markov chain Monte-Carlo method for sampling collocation points of PINNs adaptively

Yu¹,

Huang²,

Liu³

et al. 2023

Preprint

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Active Learning Based Sampling for High-Dimensional Nonlinear Partial Differential Equations

Cited by 2 publications

References 41 publications

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

MCMC-PINNs: A modified Markov chain Monte-Carlo method for sampling collocation points of PINNs adaptively

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