Journal Article

Deep Adaptive Sampling for Surrogate Modeling Without Labeled Data

Yang, Chao, Tang, Kejun, Zhai, Jiayu, Wang, Xili, Wan, Xiaoliang

2024

Overview

Surrogate modeling is of great practical significance for parametric differential equation systems. In contrast to classical numerical methods, using physics-informed deep learning-based 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 of low-regularity parametric differential equations and illustrate the necessity of adaptive sampling for constructing surrogate models. In the parametric setting, the residual loss function can be regarded as an unnormalized probability density function (PDF) of the spatial and parametric variables. In contrast to the non-parametric setting, factorized joint density models can be employed to alleviate the difficulties induced by the parametric space. The 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 through variance reduction. We demonstrate the effectiveness of the proposed method with a series of numerical experiments, including the physics-informed operator learning problem, the parametric optimal control problem with geometrical parametrization, and the parametric lid-driven 2D cavity flow problem with a continuous range of Reynolds numbers from 100 to 3200.

• Publisher: Springer US,Springer Nature B.V
• Subject: Accuracy / Adaptive sampling / Adaptive systems / Algorithms / Approximation / Cavity flow / Computational Mathematics and Numerical Analysis / Deep learning / Differential equations / Error reduction / Flight simulators / Fluid dynamics / Fluid flow / Inverse problems / Mathematical analysis / Mathematical and Computational Engineering / Mathematical and Computational Physics / Mathematical models / Mathematics / Mathematics and Statistics / Methods / Neural networks / Numerical methods / Operators (mathematics) / Optimal control / Parameterization / Partial differential equations / Physics / Probability density functions / Regularity / Reynolds number / Sampling methods / Statistical analysis / Theoretical / Two dimensional flow