State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Das, Sourav; Tesfamariam, Solomon

doi:10.48550/arxiv.2202.06416

Cited by 8 publications

(8 citation statements)

References 97 publications

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“…All statistics indicate that RANG-m achieves the best performance. Consistent with the previous result [20], the Hammersley sampling performs best among the four non-resampling methods. In this example, we can also see that RANG-m is more stable than RANG.…”

Section: Allen-cahn Equationsupporting

confidence: 88%

“…IDRLnet [40] provides jittered grid sampling. The authors of [20] explored the construction of point sets from the perspective of experimental design, compared various methods, and found that Hammersley sampling works well. In the PINN library SimNet [41], Halton pseudo-random sequence is provided as an alternative algorithm for generating low-discrepancy points.…”

Section: Hammersley Samplingmentioning

confidence: 99%

“…Therefore, a series of extensions to the vanilla PINN has been proposed to improve the accuracy and robustness of handling increasingly challenging problems. Such extensions include, but are not limited to, tuning of loss weights [11,12,13,14], novel network architectures and activation functions [15,16,17,18], domain decomposition methods [19], and mini-batch collocation generation algorithms [20,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

“…Hanna et al [39] estimated probalistic density functions to adaptively generate addtitional nodes based on the PDE residuals. From the experimental design perspective, Das and Tesfamariam [20] discussed the effects of various sampling methods and compared several node generation methods for PINN. In their experimental results, the quasi-uniform method, Hammersley sampling, seems very simple but has a significant advantage over other methods.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Allen-cahn Equationsupporting

confidence: 88%

Section: Hammersley Samplingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Wei¹,

Zhou²,

Zhang³

et al. 2022

Preprint

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“…Low-discrepancy sequences usually perform better than uniformly distributed random numbers in many applications such as numerical integration; hence, a comprehensive comparison of these methods for PINNs is required. However, very few comparisons [34,35] have been performed. In this study, we…”

Section: Related Work and Our Contributionsmentioning

confidence: 99%

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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Adaptable multi-objective optimization framework: application to metal additive manufacturing

Heddar,

Mehdi,

Matougui

et al. 2024

Int J Adv Manuf Technol

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The aim of this work is to introduce an adaptable framework for Multi-Objective Optimization (MOO) in Metal Additive Manufacturing (AM). The framework accommodates diverse design variables and objectives, enabling iterative updates via Bayesian optimization for continuous improvement. It employs space-filling design and Gaussian Process regression for high-fidelity surrogate models. A Sensitivity Analysis (SA) measures the input contributions. Multi-Objective Optimization (MOO) was performed using an evolutionary algorithm. Using literature data, the framework optimizes the surface roughness (SR) and porosity of the AM part by controlling the laser parameters. The GP model achieves cross-validation with an R² of 0.79, and with low relative mean errors. SA highlights the dominance of hatch distance in SR prediction and the balanced influence of laser speed and power on the porosity. This framework promises significant potential for the enhancement of AM technology.

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State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Cited by 8 publications

References 97 publications

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

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

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

Adaptable multi-objective optimization framework: application to metal additive manufacturing

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