BIAN: A Deep Learning Method to Solve Inverse Problems Using Only Boundary Information

Abstract: Over the past years, inverse problems in partial differential equations have garnered increasing interest among scientists and engineers. However, due to the lack of conventional stability, nonlinearity and non-convexity, these problems are quite challenging and difficult to solve. In this work, we propose a new kind of neural network to solve the coefficient identification problems with only the boundary information. In this work, three networks has been utilized as an approximator, a generator and a discriminator, respectively. This method is particularly useful in scenarios where the coefficients of interest have a complicated structure or are difficult to represent with traditional models. Comparative analysis against traditional coefficient estimation techniques demonstrates the superiority of our approach, not only handling highdimensional data and complex coefficient distributions adeptly by incorporating neural networks but also eliminating the necessity for extensive internal information due to the relationship between the energy distribution within the domain to the energy flux on the boundary. Several numerical examples have been presented to substantiate the merits of this algorithm including solving the Poisson equation and Helmholtz equation with spatially varying and piecewise uniform medium.