Solvation Free Energies from Neural Thermodynamic Integration

TL;DR

The paper addresses the challenge of computing free-energy differences between nontrivial endpoint Hamiltonians by extending NeuralTI, which learns a time-dependent neural potential $U_t$ and uses geodesic sampling to interpolate between endpoint distributions. It employs score-matching frameworks (DSM/TSM) to align the learned potential with the equilibrium potentials and uses a rolling TI estimator to evaluate $\\Delta F_{0\\to1}$ without intermediate MD/MC sampling. The approach is demonstrated on solvation systems that couple Lennard-Jones and electrostatic interactions, including LJ solvation and hydration of rigid water and methane in TIP4P water, achieving accurate free-energy estimates and showing transferability across solutes. While not claiming a speedup over traditional TI, the method offers a transferable, endpoint-only reference framework that can generalize to more complex molecular systems and potentially reduce the need for extensive intermediate sampling.

Abstract

We present a method for computing free-energy differences using thermodynamic integration with a neural network potential that interpolates between two target Hamiltonians. The interpolation is defined at the sample distribution level, and the neural network potential is optimized to match the corresponding equilibrium potential at every intermediate time-step. Once the interpolating potentials and samples are well-aligned, the free-energy difference can be estimated using (neural) thermodynamic integration. To target molecular systems, we simultaneously couple Lennard-Jones and electrostatic interactions and model the rigid-body rotation of molecules. We report accurate results for several benchmark systems: a Lennard-Jones particle in a Lennard-Jones fluid, as well as the insertion of both water and methane solutes in a water solvent at atomistic resolution using a simple three-body neural-network potential.

Figures (4)

• Figure 1: An interpolating family of distributions coupling a solute (brown) to the solvent (blue). The potentials $U_A,U_B$ and $U_{AB}$ denote the interactions within the solvent, within the solute and between the two components, respectively. The interpolation $U_t = U_A+U_B+tU_{AB}$ is often used in TI calculations to compute the free energy of the coupling of the solute, we include an additional trainable potential $t(1-t)U_t^\theta$ and train it to be the equilibrium potential at all intermediate time-slices.
• Figure 2: Unnormalized interpolating densities $e^{-\beta U_t^\theta}$ learned from different initializations. The two endpoint potentials are given by a standard Gaussian at $t=0$ ($\log Z_0 =1$) and the sum of two Gaussian densities at $t=1$ ($\log Z_1 =2$). The 4 different models learn 4 different relative weightings of the modes along the interpolation, but have a consistent prediction for the free-energy difference of $\log 2 \approx 0.69$.
• Figure 3: Solute solvation of a Lennard-Jones fluid. Left: Solvation free-energy estimates by TI and neural TI as a function of number of reference simulations and training steps, respectively. Center: Detailed view at later training stages. Softening parameters of the LJ interactions after training (right).
• Figure 4: Illustration of the coupled water (left) and methane (center) molecules. Hydration free-energy estimates of water and methane in TIP4P water (right). The dashed lines denote the experimental hydration free energies and the the curves of the same our rolling estimate from three different random seeds for each solute.