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.

翻译：本文提出一種基於Transformer的深度直接取樣方法，用於電阻抗斷層成像——一個已知的嚴重病態非線性邊界值反問題。通過學習精心設計的數據與重建影像之間的逆算子，實現了即時重建。文中著力為一個根本性問題提供具體範例：能否以及如何從數學問題的理論結構中受益，以開發任務導向且符合結構的深度神經網絡？具體而言，受反問題直接取樣方法的啟發，不同頻率下的一維邊界數據經由基於偏微分方程的特徵映射進行預處理，生成二維調和擴展作為不同輸入通道。隨後，通過引入可學習非局部核，將直接取樣重新表述為一種改進的注意力機制。該新方法在精度上優於其前身及當代算子學習方法，並在基準測試中展現出對噪聲的魯棒性。本研究將強化以下見解：儘管注意力機制最初為自然語言處理任務而設計，但它展現出極大的靈活性，可依據先驗數學知識進行修改，從而最終設計出更符合物理特性的神經架構。