Keywords: hierarchical transformer, operator learning, multiscale PDE, nested self-attention, loss function, generalization error
TL;DR: We design a hierarchical transformer based operator learning method, so that the accurate, efficient and robust computer simulation of multiscale PDE problems with an ensemble of input parameters becomes feasible.
Abstract: Complex nonlinear interplays of multiple scales give rise to many interesting physical phenomena and pose major difficulties for the computer simulation of multiscale PDE models in areas such as reservoir simulation, high frequency scattering and turbulence modeling. In this paper, we introduce a hierarchical transformer (HT-Net) scheme to efficiently learn the solution operator for multiscale PDEs. We construct a hierarchical architecture with scale adaptive interaction range, such that the features can be computed in a nested manner and with a controllable linear cost. Self-attentions over a hierarchy of levels can be used to encode and decode the multiscale solution space over all scale ranges. In addition, we adopt an empirical $H^1$ loss function to counteract the spectral bias of the neural network approximation for multiscale functions. In the numerical experiments, we demonstrate the superior performance of the HT-Net scheme compared with state-of-the-art (SOTA) methods for representative multiscale problems.
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