Deep Learning For Solving Partial Differential Equation Inverse Problems And Its Application To Magnetotelluric Inversion

Partial differential equation(PDE)inverse problems are an important area of mathematical research,and a mathematical model for many natural phenomena and engineering problems.It has a wide range of applications in modern and important engineering and scientific fields such as media imaging,image processing and remote sensing telemetry.Many traditional regularisation theories and algorithms for solving inverse problems inevitably encounter the phenomenon of “curse of dim-ensionality” when dealing with high-dimensional PDE inverse problems.It is also difficult to solve highly ill-posed problems.Deep learning inversion,as a new technique,has received much attention in recent years due to its powerful nonlinear fitting capability.This dissertation focuses on the PDE inverse problems in magnetotelluric and proposes a series of solution methods based on deep learning techniques,which are mainly as follows:Firstly,a physical information neural network(PINN)is introduced for the solution of the PDE inverse problem,the convergence principle of the PINN method is discussed,and a network training strategy based on the Adam and L-BFGS algorithms for phased optimisation is constructed.Numerical experiments are conducted using the one-dimensional Burger equation and the two-dimensional Poisson equation.The results show that the PINN training method based on the staged optimisation strategy can effectively reduce the risk of falling into local minima and improve the convergence performance and prediction accuracy of the network training.Secondly,a novel 8-layer residual neural network(Res Net1D-8)is built for 1D magnetotelluric inversion.In terms of sample production,a compact differential format algorithm is proposed for 1D forward computation,and a parallel algorithm is designed to rapidly generate a millionlevel inversion sample library.In terms of network structure,the degradation of the model with increasing depth is effectively avoided by adding shortcut connections.Batch Normalization is also added to significantly improve the training speed and generalization performance.Numerical experiments show that the Res Net1D-8 network model has the advantages of high inversion efficiency and strong generalisation capability compared to the simulated annealing algorithm.The inversion results of the measured data show that the Res Net1D-8 residual neural network model can effectively invert the subsurface electrical structure.Thirdly,an MT2DInv-Unet network model for 2D magnetotelluric inversion is constructed.In order to solve the problem that the spatial correspondence between the resistivity model and the magnetotelluric response function is difficult to establish,a deformable convolution is designed.By adding an additional offset to each sampling point of the conventional convolution operation,the network convolution operation can extract the correspondence features that exist in the target space,possessing an adaptive sensory field.And the multi-scale residual module is introduced into the network structure,which can enhance the network performance by effectively extracting the multi-scale features of the MT model and response while alleviating the problems such as gradient disappearance and network degradation.Numerical experiments show that the proposed MT2 DInvUnet inversion outperforms the currently popular fully convolutional neural network(FCN)and U-Net network.Compared with the traditional least squares regularised inversion method(LSR),the MT2DInv-Unet inversion results can effectively interpret the measured data and can obtain a reliable subsurface resistivity structure.Finally,a 3D magnetotelluric inversion scheme based on deep learning techniques is proposed,which enables end-to-end imaging from network input(observed magnetotelluric data)to output(3D resistivity model)by designing a 3D structured neural network architecture(MT3D-Net).A joint data-driven and physics-driven weighted loss function is also introduced so that the network can follow the physical constraints of the magnetotelluric data during training,thus guiding the updating of the network parameters more rationally.In addition,in order to alleviate the difficulty of producing 3D resistivity model datasets,a data augmentation method is used to increase the sample dataset capacity and improve the model generalisation capability.Numerical experiments show that the method combines the advantages of traditional inversion and data-driven inversion,which can significantly improve the stability and accuracy of magnetotelluric inversion.To this end,it is successfully applied to the inversion of synthetic models and measured magnetotelluric data,which has good application prospects.