DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations
DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations
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DOI：
10.1016/j.jcp.2022.111868
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发表时间：
2021-12
期刊：
影响因子：
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通讯作者：
Keju Tang;X. Wan;Chao Yang
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文献类型：
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作者：
Keju Tang;X. Wan;Chao Yang
本文提出了一种求解偏微分方程组的深度自适应采样(DAS-PINNS)方法，该方法用深度神经网络逼近偏微分方程组的解，并用深度生成模型产生新的配置点来精化训练集。DAS的整个过程包括两个部分：通过最小化训练集中配置点上的剩余损失来求解偏微分方程组和生成新的训练集以进一步提高当前近似解的精度。特别是，我们将残差视为一个概率密度函数，并用一种称为KRnet的深度生成模型来逼近它。来自KRnet的新样本与残差引起的分布一致，即更多的样本位于残差较大的区域，较少的样本位于残差较小的区域。类似于经典的自适应方法，如自适应有限元，KRnet充当指导训练集求精的误差指示器。与均匀分布配置点的神经网络近似算法相比，该算法可以显著提高精度，特别是对于低正则性和高维问题。通过数值实验验证了DAS-PINNS方法的有效性。
In this work we propose a deep adaptive sampling (DAS-PINNs) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to generate new collocation points to refine the training set. The overall procedure of DAS consists of two components: solving the PDEs by minimizing the residual loss on the collocation points in the training set and generating a new training set to further improve the accuracy of the current approximate solution. In particular, we treat the residual as a probability density function and approximate it with a deep generative model, called KRnet. The new samples from KRnet are consistent with the distribution induced by the residual, i.e., more samples are located in the region of large residual and less samples are located in the region of small residual. Analogous to classical adaptive methods such as the adaptive finite element, KRnet acts as an error indicator that guides the refinement of the training set. Compared to the neural network approximation obtained with uniformly distributed collocation points, the developed algorithms can significantly improve the accuracy, especially for low regularity and high-dimensional problems. We demonstrate the effectiveness of the proposed DAS-PINNs method with numerical experiments.