The accurate prediction of geometric state evolution in complex systems is critical for advancing scientific domains such as quantum chemistry and material modeling. Traditional experimental and computational methods face challenges in terms of environmental constraints and computational demands, while current deep learning approaches still fall short in terms of precision and generality. In this work, we introduce the Geometric Diffusion Bridge (GDB), a novel generative modeling framework that accurately bridges initial and target geometric states. GDB leverages a probabilistic approach to evolve geometric state distributions, employing an equivariant diffusion bridge derived by a modified version of Doob's $h$-transform for connecting geometric states. This tailored diffusion process is anchored by initial and target geometric states as fixed endpoints and governed by equivariant transition kernels. Moreover, trajectory data can be seamlessly leveraged in our GDB framework by using a chain of equivariant diffusion bridges, providing a more detailed and accurate characterization of evolution dynamics. Theoretically, we conduct a thorough examination to confirm our framework's ability to preserve joint distributions of geometric states and capability to completely model the underlying dynamics inducing trajectory distributions with negligible error. Experimental evaluations across various real-world scenarios show that GDB surpasses existing state-of-the-art approaches, opening up a new pathway for accurately bridging geometric states and tackling crucial scientific challenges with improved accuracy and applicability.

The nonlinear conjugate gradient (NLCG) algorithm is one of the popular linearized methods used to solve the frequency-domain electromagnetic (EM) geophysical inverse problem. During NLCG iterations, the model gradient guides the searching direction while the line-search algorithm determines the step length of each iteration. Normally, the line search requires solving the corresponding forward problem a few times. Since line search is usually computationally inefficient, we introduce the adaptive gradient descent (AGD) algorithm to accelerate solving the frequency-domain EM inverse problem within the linearized framework. The AGD algorithm is a variant of the classical gradient descent method and has been well-developed and widely used in deep learning. Rather than the time-consuming line search, its core idea is to algebraically manipulate the cumulative gradients and updates of the model from previous iterations to estimate the model parameter variables at the current iteration. For the inversion of magnetotelluric (MT) data, we here designed and implemented a framework using the AGD algorithm combined with the cool-down scheme to tune the regularization parameter. To improve the convergence performance of the AGD algorithm [specifying to Adam and root-mean-square propagation (RMSProp)], we proposed a tolerance strategy which has been tested numerically. To optimize the global learning rate, we carried out some comparative trials in the proposed inversion framework. The inverted results of synthetic and real-world data showed that both the AGD algorithms (Adam and RMSProp) can recover comparable results and save more than a third of CPU time compared with the NLCG algorithm.

The efficiency of solving geophysical inverse problem largely relies on the efficiency of solving the corresponding forward problem. As for electromagnetic (EM) data forward modeling in frequency domain, the conventional numerical methods, e.g. finite difference method (FDM), discretize the governing equations resulting a large linear system which is usually expensive to solve. Meanwhile, for inversion iteration we normally do not need to solve the forward problem in high precision. Thus a rapid surrogate modeling approach which uses the neural network is promising for replacing the forward modeling module in the inversion scheme. Here we proposed an algorithm which uses the neural operator to solve the EM data modeling problem in the frequency domain. To develop a surrogate model for EM data forward problem, we introduce an extended Fourier neural operator (EFNO) that enables the calculation at least 100 times faster than the conventional FDM solver while maintaining good precision. Moreover, by adding a sub-network the proposed neural operator has good generalization which has the capacity of predicting solution at any site locations and frequencies. Due to the discretization-invariance of Fourier neural operator, the neural operator trained on coarse grids can easily transfer to fine grids with only retraining part of parameters, resulting in a super-resolution prediction capability. We test our proposed method with 2-D and 3-D magnetotelluric (MT) data modeling problems, demonstrating that the EFNO has great potentials for severing as a general rapid surrogate forward solver in EM data inversion scheme.