Deep Bayesian surrogate models with adaptive online sampling for ensemble-based data assimilation

Deep learning-based surrogate models have been a promising way of dealing with the computational effort of data assimilation. Although the surrogate can reduce the computational cost, the results are influenced by the approximation error of the surrogate. Online learning methods refit surrogates to improve the accuracy using newly generated samples during iterations. However, it is still a challenge to determine which samples should be selected to refit the surrogate. In this work, we develop a Bayesian surrogate model and an online learning method to enhance the feasibility of surrogate models and the efficiency of data assimilation. First, the Bayesian surrogate model is constructed with a deep learning-based surrogate architecture and a dropout mechanism. After the training of the surrogate, the uncertainty of samples can be obtained by multiple forward inferences of the surrogate, in which the dropout is kept active. Second, the Bayesian surrogate model is combined with the ensemble smoother with multiple data assimilation (ES-MDA) algorithm to update uncertain parameters. In each iteration, an adaptive online learning method, based on the prediction uncertainty of the surrogate model, is designed to select samples for simulation and retrain the surrogate. This work provides an efficient framework to quantify the uncertainty of deep-learning surrogate models and determine the samples to retrain the surrogate. It is suitable for most deep-learning surrogate architectures and can be easily integrated into data assimilation problems. The proposed method was verified on a complex three-dimensional three-phase reservoir. The results indicated that, compared with simulation-based methods, the proposed method can achieve similar inversion results while reducing the computational cost by over 45%; compared with other surrogate-based methods, the proposed method makes the surrogate model more robust and yields the closest results to those based on numerical simulation.