Physics-Informed Deep Learning for Entropy Prediction in Heterogeneous Systems: Thermodynamic and Information-Theoretic Case Studies

Entropy production governs irreversibility and uncertainty in both physical and information-theoretic systems. While Physics-Informed Neural Networks (PINNs) successfully solve differential equations, current architectures remain inherently domain-specific. The extraction of domain-invariant entropy representations across fundamentally different physical laws remains unexplored. This paper introduces a unified Physics-Informed Deep Learning (PIDL) framework that simultaneously enforces differential equation residuals and information-theoretic bounds within a single neural architecture. We demonstrate this framework via two canonical studies: (i) a thermodynamic continuous stirred-tank reactor (CSTR) model solving governing ODEs, where a Softplus constraint strictly enforces the Second Law of Thermodynamics; and (ii) an information-theoretic financial market model solving the inverse Fokker-Planck PDE to infer latent drift and diffusion coefficients, guaranteeing diffusion positivity via a Softplus constraint while naturally inducing Shannon entropy. Three model variants are evaluated: two domain-specific baselines and one shared-encoder architecture. The PIDL framework guarantees absolute thermodynamic admissibility with zero Second-Law violations and exhibits exceptional data efficiency, retaining >90% predictive accuracy using merely 30% of available training data. Furthermore, a post-hoc Ruppeiner Riemannian geometric analysis of the learned entropy surface successfully identifies thermodynamic phase instabilities. This methodology provides a robust, domain-agnostic architecture for physics-constrained entropy modeling, advancing applications in sustainable process design and quantitative financial risk assessment.

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

Digital twin technology, when combined with physics-informed machine learning with simulation results of Aspen, offers transformative capabilities for industrial process monitoring, control, and optimization. In this work, the proposed model presents a Physics-Informed Neural Network (PINN) digital twin framework for the dynamic, tray-wise modeling of binary distillation columns operating under transient conditions. The architecture of the proposed model embeds fundamental thermodynamic constraints, including vapor-liquid equilibrium (VLE) described by modified Raoult's law, tray-level mass and energy balances, and the McCabe-Thiele graphical methodology directly into the neural network loss function via physics residual terms. The model is trained and evaluated on a high-fidelity synthetic dataset of 961 timestamped measurements spanning 8 hours of transient operation, generated in Aspen HYSYS for a binary HX/TX distillation system comprising 16 sensor streams. An adaptive loss-weighting scheme balances the data fidelity and physics consistency objectives during training. Compared to five data-driven baselines (LSTM, vanilla MLP, GRU, Transformer, DeepONet), the proposed PINN achieves an RMSE of 0.00143 for HX mole fraction prediction (R^2 = 0.9887), representing a 44.6% reduction over the best data-only baseline, while strictly satisfying thermodynamic constraints. Tray-wise temperature and composition profiles predicted under transient perturbations demonstrate that the digital twin accurately captures column dynamics including feed tray responses, reflux ratio variations, and pressure transients. These results establish the proposed PINN digital twin as a robust foundation for real-time soft sensing, model-predictive control, and anomaly detection in industrial distillation processes.