An improved differentiable finite element method for saturated–unsaturated seepage with time–stepping, parallel, and inverse strategies

Physics–informed neural networks (PINNs) integrate physical laws with the advantages of data–driven methods. However, achieving the strict precision required by PDEs is hindered in scenarios such as unsaturated seepage, not only by the non–convexity and non–uniqueness of neural networks, but also by loss balancing, time marching, and the inherent nonlinearity of the PDEs. To address these challenges, this study proposes an improved differentiable finite element method (DFEM) framework for steady and transient saturated–unsaturated seepage analysis. The governing equation is formulated in a Galerkin weak form of Richards’ equation, and the resulting residuals are embedded into a differentiable optimization framework. Thereby, the nodal pressure heads are directly optimized, neural-network parameter redundancy is avoided, and the Dirichlet boundary conditions are exactly satisfied. A vectorized implementation is developed to evaluate Gauss–point residuals in parallel and assemble the global residual efficiently within neural–network computing frameworks. A time–marching scheme is incorporated to model transient seepage evolution, respecting the structure of parabolic PDEs rather than treating time as an additional entangled network input. For inverse analysis, the DFEM solver is coupled with Bayesian optimization to identify saturated hydraulic conductivity from pressure–head observations. Numerical examples, including Poisson flow, 1D and 2D infiltration, irregular–domain seepage, layered–soil seepage, and hydraulic–conductivity identification, demonstrate that the improved DFEM achieves stable convergence and superior accuracy compared to PINN.