1,810 research outputs found
Transformer Meets Boundary Value Inverse Problems
A Transformer-based deep direct sampling method is proposed for a class of
boundary value inverse problems. 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 specific example to a
fundamental question: whether and how one can benefit from the theoretical
structure of a mathematical problem to develop task-oriented and
structure-conforming deep neural networks? Specifically, inspired by direct
sampling methods for inverse problems, the 1D boundary data in different
frequencies are preprocessed by a partial differential equation-based feature
map to yield 2D harmonic extensions as different input channels. Then, by
introducing learnable non-local kernels, the 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
Anisotropic analysis of VEM for time-harmonic Maxwell equations in inhomogeneous media with low regularity
It has been extensively studied in the literature that solving Maxwell
equations is very sensitive to the mesh structure, space conformity and
solution regularity. Roughly speaking, for almost all the methods in the
literature, optimal convergence for low-regularity solutions heavily relies on
conforming spaces and highly-regular simplicial meshes. This can be a
significant limitation for many popular methods based on polytopal meshes in
the case of inhomogeneous media, as the discontinuity of electromagnetic
parameters can lead to quite low regularity of solutions near media interfaces,
and potentially worsened by geometric singularities, making many popular
methods based on broken spaces, non-conforming or polytopal meshes particularly
challenging to apply. In this article, we present a virtual element method for
solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media
with quite arbitrary polytopal meshes, and the media interface is allowed to
have geometric singularity to cause low regularity. There are two key
novelties: (i) the proposed method is theoretically guaranteed to achieve
robust optimal convergence for solutions with merely
regularity, ; (ii) the polytopal element shape can be highly
anisotropic and shrinking, and an explicit formula is established to describe
the relationship between the shape regularity and solution regularity.
Extensive numerical experiments will be given to demonstrate the effectiveness
of the proposed method
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