[Paper Review] Multimap targeted free energy estimation

Andrea Rizzi, Paolo Carloni+1 more|arXiv (Cornell University)|Feb 15, 2023

Machine Learning in Materials ScienceMaterials Science94 references3 citations

TL;DR

This paper introduces a multimap targeted free energy perturbation (TFEP) method that accelerates quantum mechanical free energy calculations by using multiple configuration maps derived from cheap reference simulations, eliminating the need for expensive normalizing flow training. The approach achieves ~3,000× speedup over standard FEP and ~8× over prior nonequilibrium methods when computing free energy differences from force fields to DFTB3 on drug-like molecules.

ABSTRACT

We present a new method to compute free energies at a quantum mechanical (QM) level of theory from molecular simulations using cheap reference potential energy functions, such as force fields. To overcome the poor overlap between the reference and target distributions, we generalize targeted free energy perturbation (TFEP) to employ multiple configuration maps. While TFEP maps have been obtained before from an expensive training of a normalizing flow neural network (NN), our multimap estimator allows us to use the same set of QM calculations to both optimize the maps and estimate the free energy, thus removing almost completely the overhead due to training. A multimap extension of the multistate Bennett acceptance ratio estimator is also derived for cases where samples from two or more states are available. Furthermore, we propose a one-epoch learning policy that can be used to efficiently avoid overfitting when computing the loss function is expensive compared to generating data. Finally, we show how our multimap approach can be combined with enhanced sampling strategies to overcome the pervasive problem of poor convergence due to slow degrees of freedom. We test our method on the HiPen dataset of drug-like molecules and fragments, and we show that it can accelerate the calculation of the free energy difference of switching from a force field to a DFTB3 potential by about 3 orders of magnitude compared to standard FEP and by a factor of about 8 compared to previously published nonequilibrium calculations.

Motivation & Objective

• Address the challenge of poor overlap between reference (e.g., force field) and target (e.g., DFTB3) distributions in free energy calculations.
• Overcome the high computational cost of sampling slow degrees of freedom in alchemical free energy calculations.
• Eliminate the need for expensive normalizing flow training by reusing QM data for both map optimization and free energy estimation.
• Enable efficient free energy estimation using enhanced sampling and multistate estimators with multiple maps.
• Develop a one-epoch learning policy to prevent overfitting when loss evaluation is costly.

Proposed method

• Generalize targeted free energy perturbation (TFEP) to use multiple configuration maps instead of a single map.
• Train multiple maps using the same QM data used for free energy estimation, avoiding separate training overhead.
• Derive a multimap extension of the multistate Bennett acceptance ratio (MBAR) estimator for multi-state free energy estimation.
• Implement a one-epoch learning policy to minimize overfitting when loss function evaluation is expensive.
• Integrate multimap TFEP with enhanced sampling techniques (e.g., OPES) to accelerate convergence in systems with slow degrees of freedom.
• Use Z-matrix or Cartesian coordinates as input for the maps, with batch processing to improve computational efficiency.

Experimental results

Research questions

• RQ1 Can multiple configuration maps improve free energy estimation accuracy and efficiency when reference and target distributions have poor overlap?
• RQ2 Can the training overhead of normalizing flows be eliminated by reusing QM data for both map learning and free energy estimation?
• RQ3 How does the multimap TFEP method compare to standard FEP and nonequilibrium methods in terms of speed and accuracy?
• RQ4 Can the multimap approach be effectively combined with enhanced sampling to accelerate convergence in systems with slow degrees of freedom?
• RQ5 What is the impact of batch size and coordinate representation (Cartesian vs. Z-matrix) on the stability and accuracy of the free energy estimates?

Key findings

• The multimap TFEP method achieves a ~3,000× speedup in wall-clock time compared to standard alchemical free energy perturbation (FEP) when computing free energy differences from force fields to DFTB3.
• The method reduces computation by a factor of ~8 compared to previously published nonequilibrium FEP methods on the HiPen dataset.
• The multimap MBAR estimator enables robust free energy estimation using samples from multiple states, improving statistical convergence.
• Using Z-matrix coordinates with batch size 48, the method achieves a mean absolute error of ~3.8 kcal/mol on the 'good' set, with confidence intervals of ~0.04–0.08 kcal/mol.
• The one-epoch learning policy effectively prevents overfitting even when loss evaluation is expensive, maintaining stable performance across 10 replicate simulations.
• For molecules with poor convergence (e.g., 18 and 19), the method still provides estimates with large uncertainties (e.g., 3.37 kcal/mol for molecule 18), indicating limitations in sampling quality.