DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations

Tang, Kejun; Wan, Xiaoliang; Yang, Chao

doi:10.1016/j.jcp.2022.111868

Cited by 60 publications

(14 citation statements)

References 38 publications

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“…Consequently, we often need to use the information of the trained u θ and the to-be-solved PDE to construct an appropriate error indicator. For example, |Q(u θ )|, the residual of Q is a well-known candidate of error indicator which has been used in adaptive sampling method [20][21][22][23][24][25][26][27][28][29][30]. Since the calculation of the residual often involves the calculation of some derivatives and no theory guarantees the residual equivalents to the error |u − u θ |, in this paper, we will introduce some novel error indicators, in which u is approximated by a so-called empirical value which is related to u θ and the coefficients of the target PDE.…”

Section: Error Indicatorsmentioning

confidence: 99%

“…Also in 2022, the paper [24] updates the set of training points by adding sampling points to the regions where the residual is relatively large. In a very recent paper [23], a so-called DAS-PINN method uses a residual-based generative model to generate new training points for further training. On other adaptive sampling techniques, we refer to [25][26][27][28][29][30].…”

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confidence: 99%

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Adaptive trajectories sampling for solving PDEs with deep learning methods

Chen¹,

Cen²,

Zou³

2023

Preprint

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In this paper, we propose a new adaptive technique, named adaptive trajectories sampling (ATS), which is used to select training points for the numerical solution of partial differential equations (PDEs) with deep learning methods. The key feature of the ATS is that all training points are adaptively selected from trajectories that are generated according to a PDE-related stochastic process. We incorporate the ATS into three known deep learning solvers for PDEs, namely the adaptive derivative-free-loss method (ATS-DFLM), the adaptive physics-informed neural network method (ATS-PINN), and the adaptive temporal-difference method for forward-backward stochastic differential equations (ATS-FBSTD). Our numerical experiments demonstrate that the ATS remarkably improves the computational accuracy and efficiency of the original deep learning solvers for the PDEs. In particular, for some specific high-dimensional PDEs, the ATS can even improve the accuracy of the PINN by two orders of magnitude.

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Section: Error Indicatorsmentioning

confidence: 99%

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confidence: 99%

Adaptive trajectories sampling for solving PDEs with deep learning methods

Chen¹,

Cen²,

Zou³

2023

Preprint

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“…However, direct synthetic sampling from high-dimensional model input space leads to the “curse of dimensionality” (CoD) since biomass gasification performance involves numerous influential factors . A vast collection of synthetic samples may cause the PINN to fail to train . Conversely, if synthetic sampling is inadequate, then there will be significant uncertainty in the generalization ability and interpretability of the PINN model.…”

Section: Introductionmentioning

confidence: 99%

Disentangled Representation Aided Physics-Informed Neural Network for Predicting Syngas Compositions of Biomass Gasification

Ren,

Wu,

Weng

et al. 2024

Energy Fuels

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Machine learning (ML) has been extensively studied and applied in the biomass gasification field currently. However, insufficient experimental data tends to cause a mismatch between the ML model and physical mechanism, particularly for the feedstocks that do not appear in the training data set, becoming a significant challenge in creating credible ML models for biomass gasification. Therefore, this study proposes a disentangled representation-aided physics-informed neural network method, briefly called DR-PINN, to predict biomass gasification syngas components. First, the DR-PINN extracts the latent variables to represent the feedstock properties through disentangled representation learning and generates synthetic samples in the gasificationrelated latent variable space to cover a full range of feedstock types. Then, DR-PINN employs inequality constraints to embed a priori monotonic relationships into the model training loss function. Finally, experimental and synthetic samples are simultaneously considered in the model training process to realize the synergy and complementarity of actual data information and existing physical knowledge using an evolutionary algorithm. As a result, DR-PINN shows good prediction performance (the feedstocks within the training data set: R 2 ≈ 0.96, root-mean-square error (RMSE) ≈ 1.7; the feedstocks outside the training data set: R 2 ≈ 0.81, RMSE ≈ 3). Moreover, even with the feedstocks outside the training data set, the DR-PINN model can strictly abide by the prior physical monotonic relationships, with the physical consistency degree equal to 1. Overall, the proposed DR-PINN demonstrates superior generalization and interpretability compared to other methods, such as RF, GBR, SVM, ANN, and PINN.

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“…Among many neural network-based methods in the literature, the two state-of-the-art researches in solving PDE methods are the Gaussian processes regression (GPR) for PDEs [16] and the physics-informed neural networks (PINNs) [17]. Many researchers nowadays use PINNs, a mesh-free framework based on deep neural network (DNN) methods, to solve PDEs [15,[26][27][28][29]. For example, in [15], the authors use PINN to solve backward heat conduction problems, which have been long-standing computational challenges due to being ill-posed.…”

Section: Introductionmentioning

confidence: 99%

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

Mostajeran

Hosseini

2023

Math Methods in App Sciences

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In this study, we introduce a new growing neural network algorithm that is based on wavelet neural networks and call our algorithm a growing wavelet neural network (GWNN) method. We apply our proposed scheme to train a wavelet neural network to solve chemotaxis problems with blow‐up. These problems are highly nonlinear time‐dependent systems of partial differential equations, and it is a challenge to get the pattern of the solution accurately. The proposed structure is partial retraining of the network, which increases its capacity to catch the spiky pattern of the solution. Our neural network‐based algorithm allows us to solve the nonlinear chemotaxis problems without the use of linearization techniques and regularization techniques, most of which reduce the accuracy of the model. This mesh‐free‐based method can manage a variety of blow‐up models with curved boundaries without imposing an extra cost. By proving the consistency and stability of the method, we show the convergence of GWNN solutions to analytical solutions of the chemotaxis problem. Several illustrative examples and simulation results are provided to demonstrate the correctness of the results and the robust performance of the presented algorithm. Moreover, to illustrate the effectiveness of the GWNN method, we make a comparison with two other network‐based methods.

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DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations

Cited by 60 publications

References 38 publications

Adaptive trajectories sampling for solving PDEs with deep learning methods

Adaptive trajectories sampling for solving PDEs with deep learning methods

Disentangled Representation Aided Physics-Informed Neural Network for Predicting Syngas Compositions of Biomass Gasification

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

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