Collective-Variable Stein Variational Gradient Descent for Free Energy Surface Exploration
Enhanced-sampling methods enable molecular simulations to access configurational regions that are difficult to explore with conventional dynamics, but their efficiency often deteriorates as the number of collective variables (CVs) increases. Here, we introduce an enhanced-sampling framework that couples Stein variational gradient descent (SVGD) with parallel restrained molecular dynamics simulations to facilitate exploration of higher-dimensional CV spaces. SVGD collectively transports an ensemble of interacting particles toward a target probability distribution by combining mean forces that guide particles toward low-free-energy regions with kernel-mediated repulsive interactions that maintain particle diversity. We validate the approach using an analytically solvable double-well model, followed by aqueous alanine dipeptide and tripeptide in two- and four-dimensional CV spaces, respectively. In each case, the particle ensemble reproduces the corresponding target distribution, and the reconstructed free-energy surfaces agree closely with analytical or umbrella-sampling references. For alanine dipeptide and tripeptide, CV-SVGD requires substantially less aggregate MD sampling than the corresponding umbrella-sampling reference calculations. These results demonstrate the potential of CV-SVGD as a flexible, parallelizable framework for enhanced sampling in multidimensional CV spaces. We additionally provide an open-source Python implementation to facilitate its adoption.