Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning

Frank Noé, Simon Olsson, Jonas Köhler, Hao Wu (2019)
Highlighted by Jan Jensen

Figure 1A from the paper (Copyright © 2019 The Authors, some rights reserved)

The paper presents a novel method to predict free energy differences much more efficiently.  Currently, this is usually done by MD simulations and observing how often each state is visited along the trajectory. However, transitions between each state are rare, which means very long and costly trajectories, even when using various tricks to force the transitions.

The solution Noé et al. present is to "train a deep invertible neural network to learn a coordinate transformation from x to a so-called “latent” representation z, in which the low-energy configurations of different states are close to each other and can be easily sampled."

For example, the user supplies a few examples of each state [p_x(x)] and trains the NN to find a new set of variables (z) with a much simpler probability distribution [p_z(z)], by minimising the difference using the Kullback-Leibler divergence as a loss function.

Since the NN is invertible, one can now sample repeatedly from p_z(z) to get a more accurate p_x(x), which can then be reweighed to give the Boltzmann distribution. Since p_z(z) is a simple Gaussian most of the sampled structure will have a high Boltzmann probability, so you don't have to sample that many structures.

The technical main advance is the use of an invertible NN, that allow you to go both from x to z and z to x. This is done by using a NN architecture where only simple mathematically operations (addition and multiplication) that can be reversed (subtraction and division) are allowed.

It would be very interesting to see if a similar approach can be used for inverse design.


This work is licensed under a Creative Commons Attribution 4.0 International License.

Deflate to Understand Complex Molecular Kinetics

Brooke E. Husic & Frank Noé. arXiv: 1907.04101

Contributed by Jesper Madsen

Dimensionality reduction is at the core of understanding and making intuitive sense of complex dynamic phenomena in chemistry.  It is usually assumed that the slowest mode is the one of primary interest; however, it is critical to realize that this is not always so! A conceptual example hereof is a protein folding simulation (Lindorff-Larsen et al. Science 334, 517-520, 2011) where the slowest dynamical mode is not the folding itself (see Figure). What is the influence, then, of “non-slowest” modes in this process and how can it most appropriately be elucidated?

FIG: Figure 2 from the preprint: "(A) Sampled villin structures from the MD trajectory analyzed. Helical secondary structure is colored and coils are white. Each image represents five structures sampled from similar locations in TIC space as determined by a 250-center k-means model built upon the first three original TICs. The purple structure represents the folded state, and the blue structure represents the denatured state. The green structure is a rare helical misfolded state that we assert is an artifact. (B) Two-dimensional histograms for TICA transformations constructed from villin contact distances. Dashed lines indicate the regions corresponding to the sampled structures of the same color. The first TIC tracks the conversion to and from the rare artifact only. The second TIC tracks the majority of the folding process and correlates well with RMSD to the folded structure."

This work by Husic and Noé show how deflation can provide an answer to these questions. Technically speaking deflation is a collection of methods for how to modify a matrix after the largest eigenvalue is known in order to find the rest. In their provided example of the folding simulation, the dominant Time-lagged Independent Component (TIC) encapsulates the "artifact" variation that we are not really interested in. Thus, a constructed kinetic (Markov-state) model will be contaminated in several undesirable ways as discussed by the authors in great detail.  

In principle, this should be a very common problem since chemical systems have complex Hamiltonians. Perhaps the reason why we don’t see it discussed more is that ultra-rare events – real or artifact – may not usually be sampled during conventional simulations. So, with the increasing computational power available to us, and simulations approaching ever-longer timescales, this is likely something that we need to be able to handle. This preprint describes well how one can think about attacking these potential difficulties.