Physics-Informed Neural Network Digital Twin for Dynamic Tray-Wise Modeling of Distillation Columns under Transient Operating Conditions

Short-horizon forecasts of atmospheric temperature are needed to support climate-aware digital-twin systems, but such forecasts must be produced where thermal observations are incomplete. This study evaluates a physics-informed neural network for potential-temperature forecasting, constrained by a pressure-coordinate thermodynamic advection-source equation and a diabatic-source closure fit from the preceding 12-hour period and frozen before future-time training. Using hourly ERA5 reanalysis at three pressure levels, the model is evaluated as a conditional hindcast at lead times of one, two and three hours against persistence, local-trend, and two matched neural-network baselines, one of which receives the same future meteorological forcing as the PINN, helping distinguish the physical constraint from access to future forcing. In an Oklahoma development case, mean RMSE improvement over the strongest baseline grew from 8.1% at one hour to 23.8% at three hours; under an observation-density sweep down to 5% of candidate locations, this 3-hour advantage remained 14.6--16.9%, with no evidence that lower density improves performance. Under a fixed protocol transferred to an Alabama heat event with three virtual-observation layouts, three-hour improvement ranged 19.7-24.4% with consistent origin-level wins. A parallel Montana stress test, in which fixed pressure levels intersected complex terrain, produced a three-hour degradation of roughly 17.5%, identifying a terrain-related applicability limit of the formulation. Together, these results indicate that the physics constraint's benefit grows with forecast horizon, persists under severe observation sparsity, and transfers across regions, but is bounded by the validity of a fixed vertical-coordinate representation over complex terrain, evidence relevant to physics-constrained components of climate-aware forecasting and digital-twin systems.

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.

Predicting process-property relationships in manufacturing is often challenged by high experimental costs and the limited interpretability of complex 'black-box' models. This paper proposes a novel knowledge distillation framework designed to achieve high-accuracy predictions in data-scarce scenarios. The framework integrates analytical physics priors, which are systematically extracted from scientific literature via Large Language Models, into a privileged teacher model. We employ a Graph-Masked Attention layer to capture the complex physical dependencies among input variables showing strict setpoints or a combination of static and high-frequency temporal signatures. This privileged knowledge is distilled into a lightweight student predictor for inference. The feasibility and robustness of the framework are evaluated through a comprehensive experiment across five diverse manufacturing processes. To ensure statistical reliability, given the small dataset sizes, a repeated K-fold cross-validation technique is employed to quantify model stability and generalization. Results indicate that the proposed framework consistently achieves high predictive accuracy across all evaluated domains. Most importantly, the architecture demonstrates significant fault tolerance by maintaining robust predictive performance even in scenarios where LLM-derived analytical priors are suboptimal or incomplete. Furthermore, the student predictor achieves an inference frequency exceeding 6000 Hz, which facilitates real-time edge deployment on standard industrial hardware. This work provides a scalable solution for bridging the gap between theoretical physics and real-time industrial monitoring in data-limited environments.