A Physics-Informed Neural Network-based health information system for modeling circadian gut microbiome dynamics and yogurt-driven probiotic interventions

This study quantifies the influence of yogurt on gut microbiota dynamics in a circadian-modulated manner through the framework of data-driven health information modeling. Four key parameters for a set of six coupled nonlinear ordinary differential equations (ode’s) exhibiting sinusoidal 24-hour oscillations are specified. To solve this dynamical system and enable predictive analytics, we utilize a Physics-Informed Neural Network (PINN), which integrates machine learning and mechanical domain expertise into a single computational architecture. It is demonstrated that the solution is non-negative, unique, continuously dependent on the initial conditions, and exists locally and globally using Peano’s theorem, Picard–Lindelöf theorem, Gronwall’s inequality, and the invariance of non-negative odds. The PINN framework achieves relative L2 errors of less than 10−3, infers with 45× speedup over traditional ODE solvers and is resilient to 20% measurement noise. Factors derived from yogurt, particularly anti-inflammatory effect (δY) and probiotic growth characteristics (rY and KY), have the highest effect on gut ecosystem stability. Comparative simulations show that daily yogurt intake decreases inflammation by 67.0% (Cohen’s d=−569.90, p<10−130) and enhances the Microbial Balance Index (MBI) by 46.4% (p=1.2×10−12). The framework suggests the ideal time and dose for each chronotype to offer chrono-nutrition recommendations (morning: 0.5 KG at 06: 30; midday: 0.3 KG at 12: 00; and evening: 0.2 KG at 18: 00 and 21: 00). The interpretable, efficient, and scalable approach for knowledge discovery, predictive analytics, and decision support for personalized yogurts is nutrition and microbiome-based.