Targeted free energy estimation via learned mappings

Abstract

Free energy perturbation (FEP) was proposed by Zwanzig [J. Chem. Phys. 22, 1420 (1954)] more than six decades ago as a method to estimate free energy differences and has since inspired a huge body of related methods that use it as an integral building block. Being an importance sampling based estimator, however, FEP suffers from a severe limitation: the requirement of sufficient overlap between distributions. One strategy to mitigate this problem, called Targeted FEP, uses a high-dimensional mapping in configuration space to increase the overlap of the underlying distributions. Despite its potential, this method has attracted only limited attention due to the formidable challenge of formulating a tractable mapping. Here, we cast Targeted FEP as a machine learning problem in which the mapping is parameterized as a neural network that is optimized so as to increase the overlap. We develop a new model architecture that respects permutational and periodic symmetries often encountered in atomistic simulations and test our method on a fully periodic solvation system. We demonstrate that our method leads to a substantial variance reduction in free energy estimates when compared against baselines, without requiring any additional data.

Molecular simulations: past, present, and future (a Topical Issue in EPJB)

“…Such machine learning approaches help to identify the (hopefully) few degrees of freedom that capture the essential features of the process under study and that can be used to construct coarse-grained lower dimensional models amenable to human understanding [37,38]. Furthermore, generative models based on learned mappings promise to provide a way to sample the equilibrium distribution efficiently, bypassing the statistical correlations that often hamper molecular dynamics and Monte Carlo simulation [39,40]. Finally, let us mention that the relation between molecular simulation and machine learning is not necessarily a oneway street.…”

Normalizing flows for atomic solids

“…For free energy estimation in particular, flows are interesting because they do not require samples from intermediate thermodynamic states to obtain accurate estimates, unlike traditional estimators such as thermodynamic integration [2] or the multistate Bennett acceptance ratio (MBAR) method [17]. Instead, the flow model can be used as part of a targeted estimator [11,[18][19][20][21][22][23] which was demonstrated to be competitive to MBAR in terms of accuracy when applied to a small-scale solvation problem [20].…”

“…( 2). We implement the functions f k using an improved version of the model proposed by Wirnsberger et al [20]. In this model, each f k transforms elementwise either one or two coordinates of all atoms as a function of all remaining coordinates.…”

“…The spline parameters are computed as a function of the remaining coordinates using multiple layers of self-attention, a neural-network module commonly used in language modelling [27]. Wirnsberger et al [20] showed that a diffeomorphism f parameterized this way is equivariant to atom permutations, which means that permuting the input of f has the same effect as permuting the output of f . As also shown by Refs.…”

“…The above estimation method, referred to as learned free energy perturbation (LFEP) in combination with a learned model [20], is appealing, because it does not require samples from p for either training the model or for evaluating the estimator. Although Eq.…”