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.

翻译：代理建模对参数化微分方程系统具有重要的实际意义。与经典数值方法不同，基于物理信息的深度学习方法因其处理高维问题的潜力，成为构建此类系统仿真器的重要方向——该方法需要最小化随机样本训练集上的损失函数。然而，随机样本会引入统计误差，对于低正则性和高维问题的近似而言，这种误差可能成为主导误差。本文提出了一种用于代理建模的深度自适应采样方法（$\text{DAS}^2$），通过将深度自适应采样方法（DAS）[62] [Tang, Wan and Yang, 2023] 推广至低正则性参数化微分方程的代理模型构建。在参数化框架下，残差损失函数可视为空间变量和参数变量的非归一化概率密度函数（PDF）。该PDF通过深度生成模型进行近似，由此生成的新样本被添加至训练集。由于新样本符合残差诱导分布，优化后的训练集能进一步降低当前近似解中的统计误差。通过一系列数值实验（包括雷诺数连续变化范围为100至1000的参数化二维顶盖驱动方腔流动问题），我们验证了$\text{DAS}^2$方法的有效性。