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

Researchers introduce Physics-Informed Deep Learning (PIDL), a unified neural framework that enforces both differential equations and thermodynamic constraints simultaneously across different physical domains. The framework demonstrates exceptional data efficiency and zero Second Law violations in both thermodynamic and financial modeling applications.

Physics-Informed Neural Networks have transformed how scientists solve complex differential equations by embedding physical laws directly into model architectures. This research addresses a critical limitation: existing PINNs remain domain-specific, requiring separate architectures for thermodynamic versus information-theoretic systems. The PIDL framework solves this by creating a shared-encoder architecture that extracts domain-invariant entropy representations across fundamentally different physical contexts.

The dual case studies reveal the framework's versatility and robustness. In thermodynamic modeling of continuous stirred-tank reactors, Softplus constraints guarantee absolute adherence to the Second Law of Thermodynamics—eliminating entropy violations entirely. The financial market application uses inverse Fokker-Planck equations to infer hidden market parameters while maintaining diffusion positivity constraints, demonstrating the framework's applicability beyond classical physics.

Data efficiency represents the most commercially significant finding. Retaining over 90% accuracy with merely 30% of training data carries substantial implications for industries where data collection is expensive or time-consuming. The post-hoc Ruppeiner geometric analysis further validates the learned models by identifying thermodynamic phase instabilities, adding interpretability that builds confidence in predictions.

This research bridges AI and applied physics in ways relevant to sustainable process design and quantitative finance. Organizations conducting process optimization or derivative pricing could benefit from frameworks requiring fewer training examples while guaranteeing physical admissibility. The domain-agnostic architecture suggests potential applications across chemical engineering, materials science, and financial risk modeling.

• →PIDL framework achieves zero Second Law violations by embedding thermodynamic constraints directly into neural architecture design.
• →Shared-encoder architecture extracts domain-invariant entropy representations across disparate physical systems and domains.
• →Framework retains >90% accuracy using only 30% of training data, dramatically improving data efficiency for expensive experimental scenarios.
• →Post-hoc Ruppeiner geometric analysis successfully identifies thermodynamic phase instabilities from learned entropy surfaces.
• →Methodology applications extend to sustainable process design and quantitative financial risk assessment across industries.