无标签数据下用于代理建模的深度自适应采样
Deep adaptive sampling for surrogate modeling without labeled data
AI总结：
本文提出深度自适应采样方法DAS²，将残差损失视为未归一化概率密度并用深度生成模型采样以细化训练集，从而为低正则性参数微分方程构建无标签数据的代理模型，并通过参数化二维空腔流动等实验验证其有效性。
摘要：
代理建模对于参数微分方程系统具有重要的实际意义。与经典数值方法相比，使用物理信息深度学习方法为这类系统构建模拟器是一个有前景的方向，因为它具有处理高维问题的潜力，这需要在一组随机样本的训练集上最小化损失。然而，随机样本会引入统计误差，在逼近低正则性和高维问题时，这些统计误差可能成为主要误差。在这项工作中，我们提出了一种用于代理建模的深度自适应采样方法（$\ ext{DAS}^2$），其中我们将深度自适应采样（DAS）方法 [62] [Tang, Wan and Yang, 2023] 推广到为低正则性参数微分方程构建代理模型。在参数化设置下，残差损失函数可以被视为空间变量和参数变量的一个未归一化概率密度函数（PDF）。该 PDF 由一个深度生成模型来近似，从该模型中生成新样本并添加到训练集中。由于新样本与残差诱导的分布相匹配，经过细化的训练集可以进一步降低当前近似解中的统计误差。我们通过一系列数值实验证明了 $\ ext{DAS}^2$ 的有效性，其中包括雷诺数在 100 到 1000 连续范围内的参数化顶盖驱动二维空腔流动问题。
英文摘要：
Surrogate modeling is of great practical significance for parametric differential equation systems. In contrast to classical numerical methods, using physics-informed deep learning methods to construct simulators for such systems is a promising direction due to its potential to handle high dimensionality, which requires minimizing a loss over a training set of random samples. However, the random samples introduce statistical errors, which may become the dominant errors for the approximation of low-regularity and high-dimensional problems. In this work, we present a deep adaptive sampling method for surrogate modeling ($\text{DAS}^2$), where we generalize the deep adaptive sampling (DAS) method [62] [Tang, Wan and Yang, 2023] to build surrogate models for low-regularity parametric differential equations. In the parametric setting, the residual loss function can be regarded as an unnormalized probability density function (PDF) of the spatial and parametric variables. This PDF is approximated by a deep generative model, from which new samples are generated and added to the training set. Since the new samples match the residual-induced distribution, the refined training set can further reduce the statistical error in the current approximate solution. We demonstrate the effectiveness of $\text{DAS}^2$ with a series of numerical experiments, including the parametric lid-driven 2D cavity flow problem with a continuous range of Reynolds numbers from 100 to 1000.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
