Sensitivity-Constrained Neural Operators for Data-Efficient Forward and Inverse Modeling of Partial Differential Equation Systems

Neural operators provide fast surrogates for partial differential equation (PDE) solvers, but their reliability can degrade for high-dimensional spatial inputs and inverse or repeated inference. State-only training constrains solution values but not the learned input--output response. We study sensitivity-constrained neural operators (SC-NOs), which augment standard training with sampled solver-derived Jacobian supervision. Selected sensitivities from differentiable solvers or discrete adjoints are matched during training, allowing response information to be amortized across minibatches without imposing the full Jacobian at every update. We evaluate SC-NO on advection--diffusion and RANS--Spalart--Allmaras benchmarks, input-dimensionality scaling tests, long-horizon autoregressive rollout, and a shallow-water Tohoku tsunami source-inversion case. Sensitivity supervision improves forward prediction and yields larger gains in gradient-based inverse reconstruction of distributed fields. Scaling experiments show an improved accuracy--cost tradeoff for high-dimensional gridded inputs, while ablations indicate that state values and Jacobian information provide complementary supervision. In the tsunami case, SC-FNO reconstructs gridded seafloor deformation from sparse early gauge observations and forecasts subsequent wave propagation in a near-real-time proof-of-concept workflow. These results support sampled sensitivity supervision as a practical way to improve neural PDE surrogates when forward accuracy, inverse stability, robustness, and computational cost must be considered together.

Physics-Informed Neural Network Modeling of Biodegradable Contaminant Transport through GCL/SL Composite Liners

This study develops a two-domain physics-informed neural network framework for contaminant transport through a GCL/SL composite liner system, in which the thin GCL layer is treated using a steady-state advection-dispersion-biodegradation formulation and the underlying soil liner is modeled as a transient transport domain. Two formulations are evaluated against analytical and finite-element reference solutions under different leachate-head conditions: a standard PINN with soft constraint enforcement (Std-PINN) and a hard-constrained PINN (H-PINN), in which selected boundary and initial conditions are embedded directly into the trial solutions. The Std-PINN captures the overall breakthrough behavior but shows larger errors during the early transport stage, particularly under higher leachate heads where advective transport becomes more pronounced. The H-PINN reduces the optimization burden associated with penalty-based constraint enforcement and provides more accurate and stable concentration predictions, lowering the MAE from approximately 0.058-0.067 for the Std-PINN to about 0.011-0.023 for the H-PINN, while reducing the MRE from approximately 9.10%-19.16% to about 2.08%-3.14%. Parametric analyses confirm that the H-PINN with the tanh activation function and an optimized network structure provides the best predictive accuracy. The H-PINN is further extended to inverse modeling for identifying the SL degradation half-life from limited concentration observations, showing reliable convergence toward prescribed values and acceptable robustness under low-to-moderate observation noise.