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.

💡 Research Summary

In this work the authors extend the recently introduced Fokker‑Planck Score Learning (FPSL) framework from one‑dimensional to two‑dimensional collective variable (CV) spaces, enabling data‑efficient reconstruction of free‑energy landscapes from non‑equilibrium molecular dynamics (MD) trajectories. The key insight is to embed the physics of the non‑equilibrium steady state (NESS) of a periodically driven system directly into a score‑based diffusion model. By defining a diffusion‑time‑dependent effective potential that interpolates between the true tilted potential (including external forces) at τ = 0 and a uniform prior at τ = 1, the reverse‑time stochastic differential equation learns the equilibrium score ∇ ln p(x) as the gradient of a neural‑network‑parameterized energy function Uθ(x).

Training combines denoising score matching (DSM) with two optional regularization terms: (i) a time‑derivative smoothness penalty Lreg that enforces smooth variation of the learned potential with diffusion time, and (ii) a physics‑informed Fokker‑Planck regularization LFP that penalizes the residual of the NESS Fokker‑Planck equation across the whole CV domain. The latter is evaluated using uniformly sampled initial conditions, guaranteeing that even sparsely visited regions receive a physical constraint.

To respect periodicity and any known symmetries, the authors feed Fourier features (cos 2πn x/L, sin 2πn x/L) into the network. In two dimensions the full 2‑D Fourier series (cos nϕ cos mψ, cos nϕ sin mψ, …) is used, ensuring the learned potential is intrinsically periodic and eliminating the non‑local correction term that appears in the general score expression. For angular CVs, a simple transformation u = cos θ removes Jacobian singularities; diffusion is performed in u‑space and the original θ‑score is recovered via the chain rule.

The methodology is validated on three benchmark systems: (1) alanine dipeptide in water, using the backbone dihedrals ϕ and ψ; (2) a coarse‑grained lipid bilayer permeation model, with a spatial coordinate z (periodic) and an orientation angle θ (non‑periodic); and (3) an all‑atom ethanol‑through‑membrane system, again employing z and θ. In each case, short non‑equilibrium MD simulations with constant driving forces are generated, and the resulting trajectories are fed to the 2‑D FPSL pipeline.

Results show that the 2‑D FPSL reconstructions converge far faster and achieve higher accuracy than traditional one‑dimensional estimators, umbrella‑sampling with WHAM/MBAR, or grid‑based histogram methods. Hidden barriers and alternative transition pathways that are invisible in 1‑D projections become clearly resolved in the 2‑D free‑energy surfaces. When only the DSM loss is used, the learned potentials can diverge in poorly sampled regions; adding the Fokker‑Planck regularization eliminates these artifacts and yields globally consistent potentials. Fourier feature encoding dramatically reduces the number of training epochs needed for convergence, while keeping the model size modest.

Importantly, the computational cost of FPSL scales with the number of neural‑network parameters rather than with the exponential growth of grid points required for umbrella sampling in higher dimensions. The only additional expense is the longer MD sampling needed to populate the NESS distribution, which is a physical requirement rather than an algorithmic bottleneck. Consequently, the approach is readily extensible to higher‑dimensional CV spaces, making it attractive for complex biomolecular processes such as protein folding pathways, ligand‑binding funnels, or multi‑ion transport where multiple coupled coordinates are essential.

In summary, the paper presents a unified, physics‑informed, score‑based diffusion framework that overcomes the “curse of dimensionality” inherent in traditional free‑energy estimation. By leveraging NESS physics, Fourier symmetry encoding, and robust regularization, FPSL delivers data‑efficient, accurate, and scalable multidimensional free‑energy reconstructions, opening the door to routine application of high‑dimensional collective variable analyses in computational chemistry and biophysics.