[Paper Review] Data-Efficient Multidimensional Free Energy Estimation via Physics-Informed Score Learning

Daniel Nagel, Tristan Bereau|arXiv (Cornell University)|Feb 11, 2026

Protein Structure and Dynamics0 citations

TL;DR

The paper extends Fokker–Planck Score Learning (FPSL) to reconstruct 2D free-energy landscapes from non-equilibrium MD data, enabling efficient, physics-informed multidimensional estimates.

ABSTRACT

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.

Motivation & Objective

• Motivate and address the bottleneck of high-dimensional free-energy estimation in molecular systems.
• Extend FPSL to reconstruct two-dimensional free-energy landscapes from non-equilibrium trajectories.
• Incorporate physics priors and symmetries to improve robustness and accuracy in sparsely sampled regions.
• Demonstrate the approach on diverse systems including alanine dipeptide and lipid bilayer permeation to show versatility and scalability.

Proposed method

• Extend Fokker–Planck Score Learning to multidimensional CV spaces with periodic coordinates.
• Use a diffusion-model framework where the neural network learns a scalar potential U_theta, with the score s_theta derived from the NESS of a driven periodic system.
• Parameterize the score as a gradient of a learned effective potential, enforcing periodicity to simplify training.
• Incorporate regularization: (i) diffusion-time smoothness regularization or (ii) Fokker–Planck regularization to enforce physical consistency.
• Impose symmetries via Fourier features to embed periodicity and other spatial symmetries into the network input.
• Apply non-equilibrium MD data with biasing along selected CVs and marginalize to obtain 1D profiles when needed; validate against umbrella sampling baselines.

Experimental results

Research questions

• RQ1 Can FPSL be effectively extended to reconstruct two-dimensional free-energy landscapes from non-equilibrium MD data?
• RQ2 Does incorporating the NESS physics and system symmetries improve reconstruction accuracy and robustness in sparsely sampled regions?
• RQ3 How does the 2D FPSL approach compare to traditional umbrella sampling/MBAR in convergence and bias mitigation?
• RQ4 What is the impact of learning a full 2D landscape versus marginalizing to obtain 1D profiles for systems with orthogonal degrees of freedom?
• RQ5 Is FPSL agnostic to the type of collective variables and force-field resolution while remaining data-efficient?

Key findings

• FPSL can learn accurate 2D free-energy landscapes from non-equilibrium data without exponential scaling in dimensionality.
• Modeling the full 2D landscape reveals insights hidden in 1D projections and can reduce convergence issues.
• Symmetry-aware (Fourier-feature) implementations improve reconstruction accuracy and reduce variance.
• Fokker–Planck regularization enhances robustness in sparsely sampled regions, reducing unphysical extrapolations.
• The approach is validated on alanine dipeptide, coarse-grained and all-atom lipid bilayer permeation, showing data efficiency and scalability over grid-based methods.