A Spacetime RBF-Based DNNs for Solving Unsaturated Flow Problems

This study presents a novel approach for modeling unsaturated flow using deep neural networks (DNNs) integrated with spacetime radial basis functions (RBFs). Traditional methods for simulating unsaturated flow often face challenges in computational efficiency and accuracy, particularly when dealing with nonlinear soil properties and complex boundary conditions. Our proposed model emphasizes the capabilities of DNNs in identifying complex patterns and the accuracy of spacetime RBFs in modeling spatiotemporal data. The training data comprise the initial data, boundary data, and radial distances used to construct the spacetime RBFs. The innovation of this approach is that it introduces spacetime RBFs, eliminating the need to discretize the governing equation of unsaturated flow and directly providing the solution of unsaturated flow across the entire time and space domain. Various error evaluation metrics are thoroughly assessed to validate the proposed method. This study examines a case where, despite incomplete initial and boundary data and noise contamination in the available boundary data, the solution of unsaturated flow can still be accurately determined. The model achieves RMSE, MAE, and MRE values of 10−4, 10−3, and 10−4, respectively, demonstrating that the proposed method is robust for solving unsaturated flow in soils, providing insights beyond those obtainable with traditional methods.