PINN-based approach to the consolidation analysis of visco-elastic soft soil around twin tunnels

An approach based on a Physics-Informed Neural Network (PINN) is introduced to tackle the two-dimensional (2D) rheological consolidation problem in the soil surrounding twin tunnels with different cross-sections, under exponentially time-growing drainage boundary. The rheological properties of the soil are modelled using a generalized viscoelastic Voigt model. An enhanced PINN-based solution is proposed to overcome the limitation of traditional PINNs in solving integral–differential equations (IDEs) equations. In particular, two key elements are introduced. First, a normalization method is employed for the spatio-temporal coordinates, to convert the IDEs governing the consolidation problem into conditions characterized by unit-duration time and unit-area geometric domain. Second, a conversion method for integral operators containing function derivatives is devised to further transform the IDEs into a set of second-order constant-coefficient homogeneous linear partial differential equations (PDEs). By using the TensorFlow framework, a series of PINN-based models is developed, incorporating the residual adaptive sampling method to address the 2D consolidation equations of soft soils surrounding tunnels with different burial depths and cross-sections. Comparative analyses between the PINN-based solutions, and either finite element or analytical solutions highlight that the aforementioned normalization stage empowers PINNs to solve the PDEs across different spatial and temporal scales. The integral operator transformation method facilitates the utilization of PINNs for solving intricate IDEs.