Circle-Net

The study further develops a combined neural network, that integrates two subnets: one dedicated to solving the inverse problem and another to solving the forward problem within a single operator. The “subnet” refers to a small, specialized neural network as a part of a larger combined neural network architecture, where each subnet is designed to perform a specific task. Here, the forward subnet validates the interpreted outcomes of the inverse subnet, ensuring cohesive and accurate results. Integrating forward and inverse processes within a single operator arises naturally from the need to ensure that responses from interpreted results match observations, making the validation process more convenient.

The trained Circle-Net is applied to analyze a previously unseen testing dataset, which contains 5000 samples. The testing process is completed within a rapid 35.4 seconds, with each sample taking only about 7 milliseconds to interpret including inversion and response evaluation. Across the testing set, the evaluation of logarithm transmissivity results in an average coefficient of determination (R2) of 0.67 and root mean square error (RMSE) of 0.20. Similarly, the hydraulic head responses of the inverted field yields an average R2 of 0.91 and RMSE average of 0.05 m.