DAS: A deep adaptive sampling method for solving partial differential equations

Tang, Kejun; Wan, Xiaoliang; Liu, Yang

doi:10.48550/arxiv.2112.14038

Cited by 11 publications

(17 citation statements)

References 32 publications

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“…However, for high-dimensional problems, we need to use other methods, such as generative adversarial networks (GANs) [46], as was done in Ref. [41]. Moreover, the probability of sampling a point x is only considered as p(x) ∝ ε k (x) E[ε k (x)] + c. While this probability works very well in this study, it is possible that there exists another better choice.…”

Section: Discussionmentioning

confidence: 99%

“…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%

“…• Tang et al [41] (in December 2021) proposed two methods. One is a special case of RAD by choosing k = 2 and c = 0, and the other is a special case of RAR-D by choosing k = 2 and c = 0.…”

Section: Comparison With Related Workmentioning

confidence: 99%

See 3 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: Discussionmentioning

confidence: 99%

Section: Related Work and Our Contributionsmentioning

confidence: 99%

Section: Comparison With Related Workmentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…[23] found it challenging for the vanilla PINN to solve the Allen-Cahn and Hilliard equations and improved results after adopting a similar RAR strategy. Tang et al [38] introduced an additional neural network acting as an error indicator to guide the refinement, which is aimed at avoiding the large variance in uniform random sampling. Hanna et al [39] estimated probalistic density functions to adaptively generate addtitional nodes based on the PDE residuals.…”

Section: Introductionmentioning

confidence: 99%

RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks

Wei¹,

Zhou²,

Zhang³

et al. 2022

Preprint

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Learning solutions of partial differential equations (PDEs) with Physics-Informed Neural Networks (PINNs) is an attractive alternative approach to traditional solvers due to its flexibility and ease of incorporating observed data. Despite the success of PINNs in accurately solving a wide variety of PDEs, the method still requires improvements in terms of computational efficiency. One possible improvement idea is to optimize the generation of training point sets. Residual-based adaptive sampling and quasi-uniform sampling approaches have been each applied to improve the training effects of PINNs, respectively. To benefit from both methods, we propose the Residual-based Adaptive Node Generation (RANG) approach for efficient training of PINNs, which is based on a variable density nodal distribution method for RBF-FD. The method is also enhanced by a memory mechanism to further improve training stability. We conduct experiments on three linear PDEs and three nonlinear PDEs with various node generation methods, through which the accuracy and efficiency of the proposed method compared to the predominant uniform sampling approach is verified numerically.

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Adaptive Data-Driven Deep-Learning Surrogate Model for Frontal Polymerization in Dicyclopentadiene

Liu,

Abueidda,

Vyas

et al. 2024

J. Phys. Chem. B

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Frontal polymerization (FP) is a self-sustaining curing process that enables rapid and energy-efficient manufacturing of thermoset polymers and composites. Computational methods conventionally used to simulate the FP process are time-consuming, and repeating simulations are required for sensitivity analysis, uncertainty quantification, or optimization of the manufacturing process. In this work, we develop an adaptive surrogate deep-learning model for FP of dicyclopentadiene (DCPD), which predicts the evolution of temperature and degree of cure orders of magnitude faster than the finite-element method (FEM). The adaptive algorithm provides a strategy to select training samples efficiently and save computational costs by reducing the redundancy of FEMbased training samples. The adaptive algorithm calculates the residual error of the FP governing equations using automatic differentiation of the deep neural network. A probability density function expressed in terms of the residual error is used to select training samples from the Sobol sequence space. The temperature and degree of cure evolution of each training sample are obtained by a 2D FEM simulation. The adaptive method is more efficient and has a better prediction accuracy than the random sampling method. With the well-trained surrogate neural network, the FP characteristics (front speed, shape, and temperature) can be extracted quickly from the predicted temperature and degree-of-cure fields.

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DAS: A deep adaptive sampling method for solving partial differential equations

Cited by 11 publications

References 32 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

RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks

Adaptive Data-Driven Deep-Learning Surrogate Model for Frontal Polymerization in Dicyclopentadiene

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