The above procedure optimizes the force field parameters θ to reproduce the reweighted ensemble of one system. To use the reweighted ensembles of multiple systems, we use the average of the log-likelihood functions of all systems as the objective function in the optimization. We note that the objective function is a concave function of the parameters θ and ∆F provided that the potential energy function U (x; θ) is a linear function of the parameters θ. Under such conditions, the optimization problem has a unique global maximum and can be efficiently solved using gradient-based optimization algorithms. In this study, we use the L-BFGS-B algorithm to solve the optimization problem and obtain the optimized force field parameters θ.