Data-efficient multidimensional free energy estimation via physics-informed score learning
Many biological processes involve numerous coupled degrees of freedom, yet free-energy estimation is often restricted to one-dimensional profiles to mitigate the high computational cost of multidimensional sampling. In this work, we extend Fokker-Planck Score Learning (FPSL) to efficiently reconstruct two-dimensional free-energy landscapes from non-equilibrium molecular dynamics simulations using different types of collective variables. We show that explicitly modeling orthogonal degrees of freedom reveals insights hidden in one-dimensional projections at negligible computational overhead. Additionally, exploiting symmetries in the underlying landscape enhances reconstruction accuracy, while regularization techniques ensure numerical robustness in sparsely sampled regions. We validate our approach on three distinct systems: the conformational dynamics of alanine dipeptide, as well as coarse-grained and all-atom models of solute permeation through lipid bilayers. We demonstrate that, because FPSL learns a smooth score function rather than histogram-based densities, it overcomes the exponential scaling of grid-based methods, establishing it as a data-efficient and scalable tool for multidimensional free-energy estimation.
This work extends Fokker-Planck Score Learning to efficiently reconstruct two-dimensional free-energy landscapes from non-equilibrium molecular dynamics simulations using different types of collective variables, and shows that explicitly modeling orthogonal degrees of freedom reveals insights hidden in one-dimensional projections at negligible computational overhead.
@article{Nagel_2026, title={Data-efficient multidimensional free energy estimation via physics-informed score learning}, volume={165}, ISSN={1089-7690}, url={http://dx.doi.org/10.1063/5.0327436}, DOI={10.1063/5.0327436}, number={2}, journal={The Journal of Chemical Physics}, publisher={AIP Publishing}, author={Nagel, Daniel and Bereau, Tristan}, year={2026}, month=July }