In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method.

Enhancing the reliability of inversion results has always been a prominent issue in the field of geophysics. In recent years, data-driven inversion methods leveraging deep neural networks (DNNs) have gained prominence for their ability to address non-uniqueness issues and reduce computational costs compared to traditional physically model-driven methods. In this study, we propose a GMNet machine learning method, i.e., a CNN-based inversion method for gravity and magnetic field data. This method relies more on data-driven training, and in the prediction phase after the model is trained, it does not heavily depend on a priori assumptions, unlike traditional methods. By forward modeling gravity and magnetic fields, we obtain a substantial dataset to train the CNN model, enabling the direct mapping from field data to subsurface property distribution. Applying this method to synthetic data and one-field data yields promising inversion results.