Accurate prediction of tunnel seepage behavior

which is critical for estimating water inflow and ensuring construction and operational safety

is often hindered by the inherent difficulty in characterizing the formation's permeability properties. To address steady-state and transient seepage around shallow underwater tunnels

a physics-informed neural networks (PINNs) model for forward and inverse analyses was established in this study. By leveraging the automatic differentiation of neural networks to encode the governing equations

the PINNs model's loss function integrates physical constraints (governing equations

boundary conditions and initial conditions) with data mismatch from monitoring points. The effectiveness and reliability of PINNs model were validated through comparisons with benchmark solutions derived from the finite difference method (FDM). Subsequently

forward and inverse analyses were performed for steady-state and transient seepage in tunnel

considering two typical boundary conditions: constant total head and zero water pressure. The results show that in terms of forward solving

by introducing spatiotemporal coordinate affine transformation

the PINNs model can effectively simulate the transient and steady state process of tunnel seepage in anisotropic strata and achieve accurate deduction of its seepage behavior. Inversion analyses of the PINNs model can not only swiftly and accurately acquire key parameters such as the permeability coefficient

but also simultaneously deduce the seepage morphology associated with tunnel seepage. The proposed data-physics fusion PINNs model effectively reconstructs the seepage field and predicts the total head evolution around tunnel

and provides an effective way for accurate and rapid estimation of tunnel water inflow.