VESPER: Online Learning of Free Energies with a Multi-Tempered Variational Objective

Computing a molecular free-energy landscape over collective variables (CVs) is a learning problem without a dataset: the microscopic potential is known, but samples of the CV distribution must be generated by molecular dynamics (MD), which rarely crosses the barriers of interest. We introduce VESPER, which learns a neural free energy while using it to bias the simulations that produce its own training data. VESPER minimizes a single variational objective that combines several tempering levels, each flattening the landscape by a different amount. Every level is minimized by the same physical free energy, so all of them train one model together, without reweighting. At each level, biased MD configurations are compared with auxiliary walkers that explore the flattened model directly in CV space, and replica exchange between levels further accelerates exploration. We prove that the coupled sampling and learning dynamics converge under log-Sobolev and Polyak–ojasiewicz conditions. On alanine dipeptide VESPER matches or improves on established enhanced-sampling methods; it is self-consistent from three to six CVs on alanine tetrapeptide, reproduces the reference barrier for methane dehydrogenation on a nickel catalyst with a machine-learned potential, and unfolds and refolds the protein chignolin with seventeen CVs.