Neumann to Dirichlet Operator

The Neumann-to-Dirichlet (NtD) operator maps boundary conditions of one type (Neumann, specifying the flux) to another (Dirichlet, specifying the value) for partial differential equations (PDEs). Current research focuses on using this operator within machine learning frameworks, particularly deep neural networks like DeepONets, to solve inverse problems (e.g., reconstructing conductivity from electrical impedance tomography data) and efficiently handle complex PDEs, including those on graphs and with non-local terms. These advancements offer improved accuracy and computational efficiency for solving challenging problems across diverse scientific and engineering domains, such as medical imaging and material science. The development of robust and scalable algorithms for learning and applying the NtD operator is a significant area of ongoing investigation.