Bayesian neural network and Bayesian physics-informed neural network via variational inference for seismic petrophysical inversion

Deep-learning methods are being successfully applied to seismic inversion and reservoir characterization problems; however, the uncertainty quantification process has not been fully studied. We investigate probabilistic approaches for seismic petrophysical inversion by using Bayesian formulations of neural networks. In particular, Bayesian neural network via variational inference (BNN-VI) aims to estimate the probability distribution of the training parameters of the neural network, which are difficult to compute with traditional methods, by proposing a family of densities and finding the candidates that are close to the target. Variational inference (VI) has the advantage of providing a faster tool than Markov chain Monte Carlo sampling algorithms. However, for accurate predictions, BNN-VI requires large training data that are not always available in geophysical inverse problems. Hence, we develop a method that combines a Bayesian approach with physics-informed neural networks (PINNs) via VI, namely Bayesian PINNs via VI (BPINN-VI), and we apply it to a geophysical inverse problem for the estimation of petrophysical properties from seismic data. The method is based on a Bayesian network approach in which the objective function is defined by the Kullback-Leibler divergence. In addition, the physical relations between data (seismic measurements) and model variables (petrophysical properties) are embedded into neural networks through a physics-dependent loss term. We test our method on a synthetic prestack seismic data set and a real data set including an oil-saturated clastic reservoir in the North Sea with a sequence of sand and shale layers. The outcome is the most likely model of petrophysical properties. Compared with BNN-VI, the BPINN-VI results show higher correlations and R-2 coefficients, lower mean-square errors, as well as lower uncertainty and higher lateral continuity.