Title: Leveraging Physics-Informed Deep Learning to Predict Entropy in Heterogeneous Systems: Insights from Thermodynamic and Information-Theoretic Perspectives

Abstract:

Entropy generation serves as a fundamental driver of irreversibility and uncertainty across both physical and information-theoretic domains. Although Physics-Informed Neural Networks (PINNs) have proven effective for resolving differential equations, existing models are typically restricted to specific domains. The capability to extract domain-invariant entropy representations from disparate physical laws has not yet been investigated. To address this gap, this study proposes a unified Physics-Informed Deep Learning (PIDL) framework. This approach integrates differential equation residuals and information-theoretic constraints within a single neural network architecture.

We validate the framework through two distinct case studies. First, we apply it to a thermodynamic continuous stirred-tank reactor (CSTR) model, where the governing ordinary differential equations (ODEs) are solved. Here, a Softplus constraint is utilized to strictly adhere to the Second Law of Thermodynamics. Second, we employ an information-theoretic model of financial markets, which solves the inverse Fokker-Planck partial differential equation (PDE) to deduce latent drift and diffusion coefficients. In this context, the Softplus constraint ensures positive diffusion values while inherently generating Shannon entropy.

Our evaluation compares three model configurations: two domain-specific baseline models and one shared-encoder architecture. The results indicate that the PIDL framework ensures strict thermodynamic admissibility, recording zero violations of the Second Law. Additionally, the framework demonstrates superior data efficiency, maintaining predictive accuracy above 90% even when trained on only 30% of the total available data. Furthermore, a post-hoc analysis of the learned entropy surface using Ruppeiner Riemannian geometry successfully pinpointed thermodynamic phase instabilities. This methodology establishes a robust, domain-independent architecture for physics-constrained entropy modeling, with significant implications for sustainable process design and quantitative financial risk assessment.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC