Transformer Meets Boundary Value Inverse Problems
- URL: http://arxiv.org/abs/2209.14977v1
- Date: Thu, 29 Sep 2022 17:45:25 GMT
- Title: Transformer Meets Boundary Value Inverse Problems
- Authors: Ruchi Guo and Shuhao Cao and Long Chen
- Abstract summary: Transformer-based deep direct sampling method is proposed for solving a class of boundary value inverse problem.
A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and reconstructed images.
- Score: 4.165221477234755
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: A Transformer-based deep direct sampling method is proposed for solving a
class of boundary value inverse problem. A real-time reconstruction is achieved
by evaluating the learned inverse operator between carefully designed data and
the reconstructed images. An effort is made to give a case study for a
fundamental and critical question: whether and how one can benefit from the
theoretical structure of a mathematical problem to develop task-oriented and
structure-conforming deep neural network? Inspired by direct sampling methods
for inverse problems, the 1D boundary data are preprocessed by a partial
differential equation-based feature map to yield 2D harmonic extensions in
different frequency input channels. Then, by introducing learnable non-local
kernel, the approximation of direct sampling is recast to a modified attention
mechanism. The proposed method is then applied to electrical impedance
tomography, a well-known severely ill-posed nonlinear inverse problem. The new
method achieves superior accuracy over its predecessors and contemporary
operator learners, as well as shows robustness with respect to noise. This
research shall strengthen the insights that the attention mechanism, despite
being invented for natural language processing tasks, offers great flexibility
to be modified in conformity with the a priori mathematical knowledge, which
ultimately leads to the design of more physics-compatible neural architectures.
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