Mathematical Modelling of Structure-Preserving Physics-Informed Geometric Neural Stochastic Differential Equations for Molecular Dynamics

Accurate data-driven models for molecular dynamics must generate stable long-horizon predictions while providing uncertainty estimates that remain consistent with the principles of statistical mechanics. Motivated by these requirements, this study advances the comparative analysis of uncertainty quantification in molecular dynamics through a Physics-Informed Geometric Neural Stochastic Differential Equation (PI-GNSDE) framework. The proposed formulation integrates four fundamental components: an underdamped Langevin stochastic differential equation, a neural-network-based potential energy model, geometric equivariance with respect to molecular coordinates, and the fluctuation–dissipation relation, thereby enforcing thermodynamically consistent coupling between friction and thermal noise. The governing equations are presented, the mathematical assumptions required for thermodynamic consistency are established, and a reproducible workflow for data analysis and model validation is developed. Benchmark descriptors are compiled from publicly available Chignolin (CLN025) protein-folding datasets, including peptide properties, solvation conditions, thermostat and integrator configurations, simulation duration, integration time step, simulation box volume, solvent and ion atom counts, initial thermal energy, thermostat damping timescale, and total integration-step count. These quantities are reported exclusively as reproducible dataset metadata and are not interpreted as model training or predictive performance results. Furthermore, the study clearly distinguishes the mathematical formulation, modelling assumptions, numerical integration procedures, and empirical validation protocol. Since the original molecular trajectory files were unavailable during the preparation of this work, quantitative performance evaluation is intentionally omitted. Nevertheless, the proposed PI-GNSDE framework provides a mathematically rigorous, physically consistent, and reproducible foundation for uncertainty-aware molecular dynamics modelling and establishes a benchmark methodology for future comparative investigations using publicly accessible molecular simulation datasets.