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Neural networks-based variationally enhanced sampling
Luigi Bonati, Yue-Yu Zhang, Michele Parrinello
TL;DR
Standard VES represents its bias with basis functions, while sharply varying free-energy surfaces can demand flexible representations. This work uses a neural-network bias within VES and reports that the approach works across physical systems and free-energy surface dimensionalities, including a six-dimensional bias.
Problem
Standard VES expands its bias on basis functions, while sharply varying free-energy surfaces can require a more flexible representation.
Source: Neural network representation of the… · p. 3, Silicon crystallization · p. 7Method
DEEP-VES represents the bias potential as a neural network and optimizes it within the variational VES framework.
Source: Neural network representation of the… · p. 3, Conclusions · p. 8Results
Using the same architecture and similar optimization parameters, DEEP-VES handled different physical systems and free-energy surface dimensionalities, including an alanine tetrapeptide case with a six-dimensional bias.
Source: Conclusions · p. 8Takeaways & Limitations
The authors describe the work as a step toward sampling complex free-energy landscapes and identify more efficient neural-network optimization and on-the-fly collective-variable learning as future goals.
Source: Conclusions · p. 8The optimization aims for a bias close enough to the optimum to be useful rather than reaching the minimum.
Source: The optimization scheme · p. 5
Abstract
from arXiv · showhide
Sampling complex free energy surfaces is one of the main challenges of modern atomistic simulation methods. The presence of kinetic bottlenecks in such surfaces often renders a direct approach useless. A popular strategy is to identify a small number of key collective variables and to introduce a bias potential that is able to favor their fluctuations in order to accelerate sampling. Here we propose to use machine learning techniques in conjunction with the recent variationally enhanced sampling method [Valsson and Parrinello, Physical Review Letters 2014] to determine such potential. This is achieved by expressing the bias as a neural network. The parameters are determined in a reinforcement learning scheme aimed at minimizing the variationally enhanced sampling functional. This required the development of a new and more efficient minimization technique. The expressivity of neural networks allows accelerating sampling in systems with rapidly varying free energy surfaces, removing boundary effects artifacts, and making one more step towards being able to handle several collective variables.
Neural network based VES
The paper introduces DEEP-VES, replacing the standard linear basis expansion of the bias with a neural network whose parameters are optimized to minimize the VES functional.
Collective variables
- The method selects collective variables—functions of atomic coordinates whose fluctuations are critical to the process—and biases them to enhance sampling.This setup contrasts static umbrella-sampling biases with the iteratively built Gaussian bias of metadynamics.Source: Collective variables · p. 2
The variational principle
- VES minimizes a convex functional of the bias, and at its minimum the biased collective-variable distribution equals the chosen target distribution.The functional can also be expressed using the Kullback-Leibler divergence between the target and biased distributions.Source: The variational principle · p. 2
The target distribution
- Well-tempered VES targets a distribution based on the unbiased distribution, with γ > 1 regulating collective-variable fluctuations.Because the free energy is initially unknown, the target is determined self-consistently and updated every iteration in this work, though less frequent updates are possible.Source: The target distribution · p. 3
Neural network representation of the bias
- DEEP-VES represents the bias as a neural network that takes the chosen collective variables as inputs and outputs the bias potential.The network uses rectified linear unit activations in its hidden layers and a linear output layer; its continuous, differentiable representation supports force computation by back-propagation.Source: Neural network representation of the… · p. 3, Figure 1 · p. 3
The optimization scheme
- The parameters are optimized on-the-fly using stochastic gradients, in a reinforcement-learning-like scheme that treats the bias as a policy driving the system toward the target distribution.The gradient combines averages over the biased simulation and the target distribution; the latter can be estimated numerically for few collective variables or by Monte Carlo for many, avoiding exponential cost growth. The method seeks a sufficiently useful bias rather than necessarily reaching the exact minimum; the fixed bias can then support umbrella-sampling-like reweighting.Source: The optimization scheme · pp. 4–5
Results
DEEP-VES uses neural-network biases to enhance transitions and recover free-energy surfaces across suboptimal, well-chosen, and multidimensional collective variables. Reweighting refines the estimates, while the optimization schedule and parameter choices depend on the quality and dimensionality of the collective variables.
Wolfe-Quapp potential
- For the rotated Wolfe-Quapp potential using suboptimal x alone, the NN bias promoted transitions, and static reweighting produced an accurate FES without early optimization artifacts.Slowing the learning rate was important because rapid bias changes with a suboptimal CV could lead to pathways other than the lowest-energy one.Source: Wolfe-Quapp potential · p. 6
Alanine dipeptide and tetrapeptide
- Ala2 FES RMSEs were 1.5 and 0.45 kJ/mol (0.6 and 0.2 kBT), both below the chemical-accuracy threshold.The NN bias stopped updating after 12 ns; slowing or stopping learning did not affect the transition rate, indicating the chosen CVs were good.Source: Alanine dipeptide and tetrapeptide · p. 6
Silicon crystallization
- For silicon crystallization, the NN learned a useful bias despite sharp FES boundaries, enabling efficient transitions and a good reweighted FES estimate.The KL threshold was reached in 10 ns, and the bias stopped updating after 30 ns; Figure 6 shows the silicon FES, with regions above 50 kJ/mol omitted.Source: Silicon crystallization · p. 7, Figure 6 · p. 8
Choice of the parameters
- Parameter choices have limited impact with good CVs, but require more care for non-optimal CVs; the KL averaging timescale should reflect sampling of the target distribution.The protocol was robust for KL threshold ε around 0.5; larger values may suit higher-dimensional FESs, while a lower learning-rate decay time helps reach a constant bias sooner.Source: Choice of the parameters · pp. 7–8
Alanine dipeptide and tetrapeptide
- For six-dimensional Ala4, DEEP-VES used Monte Carlo estimation and reweighting to obtain FES profiles, with reference comparisons reported against metadynamics and Parallel Bias metadynamics.The supplied Figure 5 shows reweighted surfaces as functions of three dihedral angles; comparisons and other angle pairs are in the SI.Source: Alanine dipeptide and tetrapeptide · pp. 6–7, Figure 5 · p. 7
Conclusions
DEEP-VES uses neural networks to represent the bias potential and free-energy surface within VES, handling different systems and dimensionalities with the same architecture and similar optimization parameters. Its variational learning scheme and new minimization procedure support sampling complex, high-dimensional free-energy landscapes.
- With the same architecture and similar optimization parameters, DEEP-VES handled different physical systems and free-energy-surface dimensionalities, including a 6-dimensional Alanine Tetrapeptide bias and cases with an important degree of freedom omitted.The authors plan to extend the approach to even higher-dimensional landscapes.Source: Conclusions · p. 8
- The neural network is optimized by a variational principle, while the target distribution enables efficient sampling of relevant configurational space, particularly for high-dimensional free-energy surfaces.Source: Conclusions · p. 8
- The authors developed an alternative to ref. that globally tempers the bias using the Kullback–Leibler divergence between current and target distributions.They suggest conventional VES may also benefit; future goals include more efficient neural-network optimization and learning collective variables on-the-fly.Source: Conclusions · p. 8
Materials and methods
The methods implement VES-NN with PLUMED and LibTorch, then specify simulation and optimization setups for the Wolfe–Quapp potential, peptides, and silicon.
Materials and methods
- VES-NN is implemented in modified PLUMED2 [38] linked to LibTorch, with PLUMED computing parameter gradients and sampling and numerical integration supplying the functional expectations.The second expectation is evaluated by grid integration or Monte Carlo; code release in PLUMED was planned for the future.Source: Materials and methods · p. 9
Wolfe-Quapp potential
- Wolfe–Quapp simulations used a rotated potential, Langevin dynamics (timestep 0.005, target temperature 1), and a [48,24,12]-node bias network updated every 500 steps.The learning rate was 0.001; the well-tempered bias factor was 10, the integration grid had 100 bins, and the running-KL threshold was 0.5.Source: Wolfe-Quapp potential · p. 9
Peptides
- Peptide simulations used GROMACS patched with PLUMED, the Amber99-SB force field, NVT conditions, a 2-fs timestep, and a 300K target temperature.Ala2 used a 50x50-bin grid; Ala4 used Metropolis sampling of 25,000 target-distribution points per iteration and reduced its learning rate after 10 ns.Source: Peptides · p. 9
Silicon
- Silicon simulations used a 216-atom Stillinger–Weber supercell in NPT conditions, with cubic-diamond atom count as the collective variable and a bias factor of 100.The thermostat target was 1700K, the timestep was 2 fs, and the integration grid had 216 bins.Source: Silicon · p. 9