Inference of Onsager coefficient from microscopic simulations by machine learning

Dynamic density functional theory (DDFT) is a fruitful approach for modeling polymer dynamics, benefiting from its multiscale and hybrid nature. However, the Onsager coefficient, the only free parameter in DDFT, is primarily derived empirically, limiting the accuracy and broad application of DDFT. Herein, we propose a machine learning-based, bottom-up workflow to directly extract the Onsager coefficient from molecular simulations, circumventing partly heuristic assumptions in traditional approaches. In this workflow, the Onsager coefficient is derived from the proposed DDFT-informed ordinary differential equation network, trained to replicate density evolution observed in Brownian dynamics (BD) simulations. We validate our method by studying the lamellar transition in symmetric diblock copolymer melts, where the DDFT model with the extracted Onsager coefficient precisely reproduces both the density evolution and interface narrowing predicted by BD simulations, thereby demonstrating the reliability of the present scheme. Meanwhile, our studies reveal the strong relevance of the Onsager coefficient with dynamic processes and identify the explicit connection between dynamic correlations, characterized by the correlation strength and correlation length, and the system parameters, e.g., the Flory-Huggins interaction parameter. We found that far from the transition point, the correlation that transmits the thermodynamic force into a density current is localized and strong, while close to the transition point, it becomes long-ranged but weak. Our approach aims to develop a more generalized framework to bridge more refined particle-based simulations to more coarse-grained field-based calculations, and the insights gained by using our approach could be extended to other non-equilibrium systems in polymer sciences.