Theory-guided Auto-Encoder for surrogate construction and inverse modeling

A Theory-guided Auto-Encoder (TgAE) framework is proposed for surrogate construction, and is further used for uncertainty quantification and inverse modeling tasks. The framework is built based on the Auto-Encoder (or Encoder–Decoder) architecture of the convolutional neural network (CNN) via a theory-guided training process. In order to incorporate physical constraints for achieving theory-guided training, the governing equations of the studied problems can be discretized by the finite difference scheme, and then be embedded into the training of the CNN. The residual of the discretized governing equations, as well as the data mismatch, constitute the loss function of the TgAE. The trained TgAE can be utilized to construct a surrogate that approximates the relationship between the model parameters and model responses with limited labeled data. Several subsurface flow cases are designed to test the performance of the TgAE. The results demonstrate that satisfactory accuracy for surrogate modeling and higher efficiency for uncertainty quantification tasks can be achieved with the TgAE. The TgAE also shows good extrapolation ability for cases with different correlation lengths and variances. Furthermore, inverse modeling tasks are also implemented with the TgAE surrogate, and satisfactory results are obtained.