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 PINNbased 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.

This study investigates the rheological properties of saturated soft clay surrounding a tunnel using the generalized Voigt viscoelastic model. The model incorporates linear semi-permeability boundary conditions to describe the behavior of the clay. Furthermore, two-dimensional rheological consolidation control equations are derived based on the Terzaghi-Rendulic theory, considering the excess pore water pressure as a variable. To solve the equations, conformal transformation and separation of variables methods are employed, resulting in two independent equations representing the excess pore pressure in terms of time and space variables. The Laplace transformation and partial fractional summation method are then utilized to obtain the solution for excess pore pressure dissipation in the time domain. The reliability of the solution is verified by comparing it with the existing four-element Burgers and five-element model, both of which are derived from the generalized Voigt model. Furthermore, the influence of liner permeability, Kelvin body number, independent Newtonian dashpot viscosity coefficient, and tunnel depth on the dissipation and distribution of excess pore pressure is analyzed based on the established solutions. The findings indicate that a higher relative permeability of the liner and soil leads to an earlier onset of excess pore pressure dissipation and a faster dissipation rate. Increasing the number of Kelvin bodies results in slower dissipation rate. Moreover, larger independent viscous coefficients lead to smaller viscous deformation and faster dissipation rates. Additionally, greater tunnel depth prolongs soil percolation path, slowing down the dissipation of excess pore pressure. When the relative permeability coefficient is 0.01, the excess pore pressure gradually decreases with distance from the outer wall of the tunnel. However, when the relative permeability coefficient is 1, the excess pore pressure initially increases and then decreases with distance. As the relative permeability coefficient increases, the influence of the number of Kelvin bodies on the dissipation of super pore pressure diminishes, the variation in super pore pressure dissipation caused by different independent Newtonian dashpot viscosity coefficients gradually decreases, and the role of tunnel liners as new permeable boundaries within the soil layer is becoming increasingly prominent.