Deep adaptive sampling for surrogate modeling without labeled data
CoRR(2024)
Abstract
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 (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 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.
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