A multi-output constitutive physics-informed neural network (MOC-PINN) for seepage problems with free surface

The main challenge in seepage problems with free surface is that the free-surface boundary is unknown before computation and needs to be determined iteratively. Although physics-informed neural networks (PINNs) offer a mesh-free advantage, their application to such problems often requires explicit iteration and higher-order automatic differentiation, which limits efficiency. To address this problem, this paper proposes a multi-output constitutive physics-informed neural network (MOC-PINN) for free-surface seepage problems. This method employs a network architecture capable of handling heterogeneous media and simultaneously predicting hydraulic head and seepage velocity. By extending seepage governing equations to the full domain and introducing the Heaviside function combined with Darcy’s law to establish intrinsic constitutive constraints among multi-output variables, it constructs a composite loss function, enabling the distinction of the spatial distribution of dry and saturated regions without explicit continuity conditions at the dry-saturated interface. This enables implicit capture of the free surface during training and avoids complex interface tracking. Furthermore, by directly outputting the seepage velocity and substituting it into the governing equations, the method circumvents second-order automatic differentiation of the hydraulic head field, reducing computational cost. Numerical results indicate that the proposed algorithm achieves over 22% speedup and 9.1% accuracy improvement under fine collocation compared with iterative PINN methods, and shows more robust convergence in complex scenarios. A proof-of-concept inverse example further demonstrates the framework’s extensibility to parameter identification, demonstrating its practical value and extensibility.