The main challenge in the inversion of seismic data to predict the petrophysical properties of hydrocarbon-saturated rocks is that the physical relations that link the data to the model properties often are nonlinear and the solution of the inverse problem is generally not unique. As a possible alternative to traditional stochastic optimization methods, we develop a method to adopt machine-learning algorithms by estimating relations between data and unknown variables from a training data set with limited computational cost. We develop a probabilistic approach for seismic petrophysical inversion based on physics-informed neural network (PINN) with a reparameterization network. The novelty of our approach includes the definition of a PINN algorithm in a probabilistic setting, the use of an additional neural network (NN) for rock-physics model hyperparameter estimation, and the implementation of approximate Bayesian computation to quantify the model uncertainty. The reparameterization network allows us to include unknown model parameters, such as rock-physics model hyperparameters. Our method predicts the most likely model of petrophysical variables based on the input seismic data set and the training data set and provides a quantification of the uncertainty of the model. The method is scalable and can be adapted to various geophysical inverse problems. We test the inversion on a North Sea data set with poststack and prestack data to obtain the prediction of petrophysical properties. Compared with regular NNs, the predictions of our method indicate higher accuracy in the predicted results and allow us to quantify the posterior uncertainty.

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