Physics-Informed Neural Network Modeling of Biodegradable Contaminant Transport through GCL/SL Composite Liners
Authors: Dong Li, Yapeng Cao, Haiping Zhao, Shutong Han
Organizations: PhD, Department of Civil, Environmental, and Infrastructure Engineering, George Mason University, Fairfax, VA 22030, USA · 2* PhD, State Key Laboratory of Cryospheric Science and Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou 730000, China · Laboratoire Navier/CERMES, École Nationale des Ponts et Chaussées, Institut Polytechnique de Paris, 77455 Marne-la-Vallée cedex 2, France · 3MS, Department of Civil and Environmental Engineering, the University of New South Wales, Sydney, Australia
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.
While physics-informed neural networks (PINNs) have shown strong potential for process modeling, physical equations are only enforced as soft constraints during training, and thus, they do not guarantee constraint satisfaction at inference. We propose a framework, called piecewise-linear Karush--Kuhn--Tucker hard-constrained PINNs (PL-KKT-hPINNs), that strictly enforces nonlinear equality constraints through piecewise-linear projection. This extends the KKT-hPINN framewor, which exactly enforces linear equalities through the Karush--Kuhn--Tucker (KKT) conditions associated with orthogonally projecting neural network outputs onto the constraint feasible region. The method is demonstrated on a continuous stirred-tank reactor (CSTR) case study for both one and two inputs. Results show that PL-KKT-hPINN preserves predictive accuracy comparable to that of a standard neural network while achieving substantially lower constraint violations. In addition, the proposed model shows improved robustness in low-data regimes, yielding lower RMSE than the unconstrained neural network for limited training sample sizes. These results demonstrate that PL-KKT-hPINN provides a computationally efficient and physically consistent framework for surrogate modeling of nonlinear chemical engineering systems.
Fateme Mohammad Mohammadi, Hector Budman, Joshua L. Pulsipher
We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.
This study develops a dimensionless multi-domain physics-informed neural network (PINN) framework for electro-osmotic radial consolidation considering smear effects and combined vacuum and surcharge loading. Three PINN-based models are investigated: a standard soft-constrained PINN (Std-PINN), a modified gated PINN (Mod-PINN), and a modified gated PINN with hard-constraint boundary encoding (Mod-HC-PINN). The models are evaluated against FEM reference solutions under four loading cases, including constant vacuum, exponential vacuum, exponential vacuum with ramp surcharge, and exponential vacuum with cyclic haversine surcharge. The results indicate that the gated architecture applied in Mod-PINN improves the resolution of steep pressure gradients near the cathode and smear-zone interface under constant vacuum loading. Under time-dependent loading, the soft-constrained Mod-PINN shows reduced accuracy because it must learn multiple competing objectives simultaneously. The Mod-HC-PINN mitigates this issue by embedding the cathode boundary and initial conditions into the output structure, thereby reducing the optimization burden and improving physical consistency. The Mod-HC-PINN achieves MAE values of 0.43, 0.41, and 0.27 kPa for the exponential vacuum, ramp surcharge, and cyclic surcharge cases, respectively. Sensitivity analyses further demonstrate that the proposed framework remains robust across practical ranges of network architecture, collocation density, and permeability contrast.