Solving the Maxwell’s Equations From Magnetic Dipole Sources in 2.5-D TI Medium With PINNs

This article introduces a novel physics-informed neural networks (PINNs) to solve Maxwell’s equations with triaxial magnetic dipole sources in transverse isotropy (TI) medium, which can be applied to simulate the tool response of electromagnetic (EM) wave well logging. The physical model consists of a planar-stratified formation and some cylindrical, axial symmetrical anomalous bodies embedded it. The EM field is a 3-D distribution, especially for horizontal magnetic dipoles. By analyzing the characteristics of the field, the azimuth angle $\phi $ is extracted from the initial partial differential equations (PDEs); thus, the task is transferred into a 2-D issue about input parameters $\rho $ and z. The singularity of dipole sources is addressed well by abstracting the background field. Each field component includes both the real part and imaginary part in a dissipative medium. Therefore, we train two sets of PINNs to solve for them, respectively, which greatly increases the accuracy of output results. The lost functions related to the soft boundary and hard boundary are deeply discussed, which leads to different accuracy and efficiency. Finally, we simulate the magnetic field signals for some typical well logging scenarios in vertical and deviated wells. The results are validated by comparing with mature analytical solutions and the finite element method (FEM). The proposed PINNs method provides an innovative approach for forward modeling of EM wave logging for dealing with local anomalous body (2.5-D).