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arXiv 2209.14977cs.LGcs.NAmath.NA
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Transformer满足边值反问题

Transformer Meets Boundary Value Inverse Problems

University of California, Irvine(加州大学欧文分校);University of Missouri-Kansas City(密苏里大学堪萨斯城分校)

Ruchi Guo, Shuhao Cao, Long Chen

摘要:

针对电阻抗成像这一严重不适定的非线性边值反问题,提出了一种基于变换器的深度直接采样方法。通过评估精心设计的数据和重建图像之间的学习逆算子来实现实时重建。一个努力是给一个具体的例子,一个基本的问题:是否以及如何可以受益于数学问题的理论结构,以开发面向任务和结构符合深神经网络?具体而言,受逆问题的直接采样方法的启发,通过基于偏微分方程的特征映射对不同频率的1D边界数据进行预处理,以产生2D谐波扩展作为不同的输入通道。然后,通过引入可学习的非局部核函数,将直接采样转换为一种改进的注意力机制。新方法实现了优越的精度比其前辈和当代运营商学习,并显示出对基准噪声的鲁棒性。这项研究将加强的见解,尽管被发明的自然语言处理任务,注意力机制提供了很大的灵活性,以修改符合先验数学知识,这最终导致更多的物理兼容的神经架构的设计。

英文摘要:

A Transformer-based deep direct sampling method is proposed for electrical impedance tomography, a well-known severely ill-posed nonlinear 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 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 new method achieves superior accuracy over its predecessors and contemporary operator learners and shows robustness to noises in benchmarks. This research shall strengthen the insights that, despite being invented for natural language processing tasks, the attention mechanism 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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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