Physics-guided deep-learning direct current-resistivity inversion with uncertainty quantification

Electrical resistivity tomography (ERT) is a nonlinear and ill-posed inverse problem, typically addressed through deterministic methods. These approaches are prone to converge to a local minimum solution and do not provide accurate uncertainty quantification. Standard probabilistic approaches overcome these limitations but are computationally expensive. We develop a computationally efficient, physics-guided deep-learning approach to ERT inversion that accounts for uncertainty estimation. Our strategy integrates deep learning with the discrete cosine transform to reduce the number of model parameters, thereby speeding up the training phase and reducing the ill-conditioning of the ERT problem. To enable the network to learn by minimizing the data misfit, a forward-modeling process is incorporated into the network training, thus projecting the predicted model onto the predicted data. Then, a Monte Carlo dropout (MCD) approach is used to estimate the uncertainty affecting the solution. We demonstrate the applicability of our method using synthetic and field data and by comparing the results with the outcomes of a standard deterministic inversion and supervised learning. We also validate the model uncertainties provided by the MCD approach with those obtained from a gradient-based Markov chain Monte Carlo inversion. Our findings indicate that the physics-guided approach outperforms supervised learning and achieves data fitting similar to that offered by classical deterministic inversion, with the advantage of providing accurate model uncertainties with a negligible amount of extra computational effort.