Session B26: Predicting Rare Event Kinetics in Complex Systems with Theory, Simulations and Machine Learning I

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Boltzmann Generators are a novel rare-event sampling framework that combines invertible neural networks (normalizing flows) with statistical-mechanics based reweighting methods [1]. The basic idea is to train the invertible neural network to generate samples from a probability distribution similar to the target equilibrium distribution (e.g. canonical), and then reweight to the exact target density so as to generate asymptotically unbiased samples. Ways to incorporate symmetries of the energy function [2] and stochastic sampling steps in the neural network [3] have also been proposed. In this talk I will describe recent methodological developments and applications to the sampling of classical many-body systems.


[1] https://science.sciencemag.org/content/365/6457/eaaw1147
[2] https://arxiv.org/abs/2006.02425
[3] https://arxiv.org/abs/2002.06707   

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We have developed molecular latent space simulators (LSS) to learn highly efficient and accurate surrogate models of molecular dynamics (MD) by stacking three specialized deep learning networks to (i) encode a molecular system into a slow latent space, (ii) propagate dynamics in this latent space, and (iii) generatively decode a synthetic molecular trajectory. The trained LSS generates novel ultra-long molecular trajectories at six orders of magnitude lower cost than MD enabling resolution of rare thermodynamic states and kinetic transitions with arbitrarily low statistical uncertainties. In an application to Trp-cage, we generate millisecond trajectories in just minutes of wall clock time and demonstrate excellent agreement with the MD structure, thermodynamics, and kinetics.

H. Sidky, W. Chen, and A.L. Ferguson Chem. Sci. 11 9459 (2020)   

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Deep neural networks, when optimized with sufficient data, provide accurate representations of high-dimensional functions; in contrast, function approximation techniques that have predominated in scientific computing do not scale well with dimensionality. While the promise of unparalleled accuracy may suggest a renaissance for applications that require parameterizing representations of complex systems, in many applications gathering sufficient data to develop such a representation remains a significant challenge. Here we introduce an approach that combines rare events sampling techniques with neural network optimization to optimize objective functions that are dominated by rare events. We show that importance sampling reduces the asymptotic variance of the solution to a learning problem, suggesting benefits for generalization. We study our algorithm for identifying dynamical transition pathways between two states of a system, a problem with applications in statistical and chemical physics. Our numerical experiments demonstrate convergence even with the compounding difficulties of high-dimension and rare data.   

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The rapid advances in computational power have made molecular dynamics (MD) a powerful tool for studying biophysical systems. However, there are still at least two open questions in this area: first, how to make use of the deluge of data generated from MD simulation human understandable; second, how to further push the limit of the timescale that can be reached by MD. One key to both difficulties is to uncover a low dimensional manifold (known as reaction coordinate or RC) on which the dynamics of the system can be projected. Here we developed State Predictive Information Bottleneck (SPIB) to learn RC from trajectories. In SPIB, the time delay is interpreted as the minimal time resolution that we care for a dynamic system, and can be used to automatically discretize the high-dimensional state space into a few metastable states. Such a discrete-state representation then can be employed to guide our RC to focus only on the motion related to the state transitions. In this way, the RC learned by SPIB is strongly related to the committor, and is able to identify the correct transition states.   

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Nonequilibrium forces often regulate how quickly a complex random system transitions between long-lived states, but there are few theories or guiding principles for how transition rates are enhanced by time-dependent external influence. Using tools from stochastic thermodynamics, we develop a general theory of rate enhancement in the transition path ensemble, leading to fundamental bounds on the ratio of transition rates. Under generic physical conditions, we show that the heat dissipated over the course of the transition sets an upper limit on achievable rate enhancement. This basic tradeoff between speed and energy consumption is illustrated in canonical examples of barrier crossing under autonomous, active, and periodically modulated driving protocols. We discuss how our bounds can be employed as a variational principle to design optimal protocols for tuning a system to achieve a target rate, and inferring rare reaction rates in (non)equilibrium settings.   

Live

In a rare event scenario, a system undergoes transitions between metastable states separated by large barriers. Calculating transition rate under this circumstances is difficult. However, it has been shown that if a bias can be engineered such that the transition state region is left untouched, transition rates can be nonetheless computed[1-3]. Here, we present a new method to build a bias that achieves this property in a relatively simple and transparent way. In this method, called Gaussian Mixture Based Enhanced Sampling, the metastable states are discriminated by a set of descriptors[4]. The fluctuations of these descriptors in the different metastable states are fitted to Gaussian Mixtures. From the fit, a bias can be built such that it allows the system to migrate from one metastable state to another. Yet, by construction, the bias is null in the transition state region. We exemplify the effectiveness of the method by calculating rates for chemical reaction, ligand unbinding and protein unfolding.
1.Voter,A. F., J. Chem. Phys., 1997, 106, 4665-4677
2.Tiwary,P.; Parrinello,M., Phys. Rev. Lett. 2013, 111, 230602
3.McCarty,J. et. al., Phys. Rev. Lett. 2015, 115, 070601
4.Debnath, J.; Parrinello M., J. Phys. Chem. Lett. 2020,11(13), 5076-5080   

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Phase transitions are ubiquitous in nature and play a critical role in a broad range of phenomena from ice
formation in clouds to signaling in biological cells. Often the phase transitions include molecular length
and timescales that are hard to probe in experiments but conducive to the lengthscales accessible in
molecular simulations. However, the timescales to observe these phase transitions can be beyond the
accessible timescales in simulations. Consequently, advanced sampling methods are required to sample
such transitions as well as to extract the key information from those sampling.

One such phase transition of interest in our research is liquid-to-solid transition. We use path sampling
methods to sample these transitions – specifically we use forward flux sampling and transition interface
sampling. While the sampling can be achieve several challenges remain – (i) how do we identify the
reaction coordinates (i.e. combination of order parameters) that best ascertain the direction of the
transition (ii) how do we identify the various structures (known and possibly unknown) that form during
such transitions. To this end, we capitalize on the benefits of artificial intelligence methods. Specifically,
we have used PointNet based approaches to identify various structures. We will also discuss our efforts
for using machine learning based protocols for discovering reaction coordinates from the pathways
sampled in our simulations.   

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Recurrent neural networks (RNNs) have led to breakthroughs in natural language processing and speech recognition. Here we show that RNNs, specifically long short-term memory (LSTM) neural networks can also capture the temporal evolution of chemical/biophysical trajectories. Our language model learns a probabilistic model of 1-dimensional stochastic trajectories generated from higher-dimensional molecular dynamics. The model captures the Boltzmann statistics of the system and also reproduces kinetics across a large spectrum of timescales. We demonstrate how training the LSTM is equivalent to learning a path entropy, and that the LSTM embedding layer, instead of representing contextual meaning of characters, here exhibits a nontrivial connectivity between different metastable states in the underlying physical system. We also demonstrate our model's reliability through different benchmark systems and a single molecule force spectroscopy trajectory for multi-state riboswitch.   

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Syndiotatic polysterene (sPS) exhibits complex polymorphic behaviour, resulting in rugged free energy landscapes (FELs) with high energy barriers. Enhanced sampling methods have the potential remedy to overcome the barriers with prior knowledge of collective variables (CVs), typically identified through physical or chemical expertise. Autoencoders are powerful tools for providing a low-dimensional embedding of the essential features, since the technique forces an information compression in the bottleneck region. A specialised autoencoder architecture, the Gaussian mixture variational autoencoder (GMVAE), performs dimensionality reduction and clustering within a single unified framework, and can identify the inherent dimensionality of the system by enforcing physical constraints in the latent space. In order to efficiently describe the local environment of sPS monomers, we adapt an atomic representation used in machine learning. One of the advantages of using these descriptors is that they do not require incorporation of excessive system-specific intuition and demonstrate good transferability properties. With this data-driven approach, we aim to characterise the pathways between polymorphic transitions.