A fast uncertainty TEM inversion method for non-fixed-length data based on mixture density network

Abstract The time-domain electromagnetic method (TEM) is a widely used earth exploration approach for underground resource exploration and engineering investigations. In practical applications, engineers aim to obtain accurate information about subsurface media on-site to rapidly and reliably assess underground structural conditions. However, due to the complexity of geological structures and the multi-solution problem of inversion, there is a lack of efficient inversion methods capable of producing accurate representations of subsurface structures, particularly when applying them to the non-fixed-length TEM response. This paper proposes a fast uncertainty-estimation inversion technique for non-fixed-length TEM data based on mixture density network to solve this problem. This method performs rapid inversion by using a neural network forward pass to directly predict the parameters of a Gaussian-mixture posterior distribution, thereby avoiding conventional Bayesian sampling–based inference. Moreover, because it outputs a full probability distribution rather than a single point estimate, it enables quantitative uncertainty assessment through metrics such as credible intervals. Field experiments demonstrate that inverting data from 194 survey points requires only 9.405 ms, with an average relative error of 0.0150 for the maximum a posteriori model. This proposed method not only quantitatively evaluates the uncertainty of inversion results but also handles non-fixed-length TEM responses. Additionally, it significantly reduces the time required for inversion and evaluation, providing a new solution for obtaining accurate TEM inversion results in real-time at field construction sites.