Abstract: 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 combines 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, which enforces a thermodynamically consistent coupling between friction and thermal noise.
The paper presents the governing equations, explicitly states the assumptions required to guarantee thermodynamic consistency, and establishes a reproducible workflow for data analysis and model validation. To ensure transparency and traceability, benchmark descriptors are compiled from publicly available Chignolin (CLN025) protein-folding datasets, including peptide properties, solvation conditions, thermostat and integrator configurations, the number and duration of simulations, integration time step, simulation box volume, solvent and ion atom count, initial thermal energy, thermostat damping timescale, and total integration-step count. These quantities are reported exclusively as reproducible dataset metadata and should not be 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 raw molecular trajectory files were not available during the preparation of this work, quantitative performance measures, including Kullback-Leibler divergence, energy drift, prediction coverage, and long-term stability metrics, are intentionally excluded to avoid unsupported conclusions. Nevertheless, the proposed framework provides a rigorous, transparent, and reproducible foundation for future model training, calibration, uncertainty quantification, and validation once the corresponding trajectory data are acquired and processed using the prescribed pipeline.
Keywords: Molecular dynamics; mathematical modelling; stochastic differential equations; uncertainty quantification; Langevin dynamics; geometric deep learning; neural stochastic differential equations; fluctuation-dissipation theorem; BAOAB integration; Chignolin.
Title: Mathematical Modelling of Structure-Preserving Physics-Informed Geometric Neural Stochastic Differential Equations for Molecular Dynamics
Author: Jeremiah U. Atsu, Samuel O. Essang, Jackson E. Ante
International Journal of Mathematics and Physical Sciences Research
ISSN 2348-5736 (Online)
Vol. 14, Issue 1, April 2026 - September 2026
Page No: 121-134
Research Publish Journals
Website: www.researchpublish.com
Published Date: 04-August-2026
