For a transition between two stable states, the committor is the probability that the dynamics leads to one stable state before the other. It can be estimated from trajectory data by minimizing an expression for the transition rate that depends on a lag time. We show that an existing such expression is minimized by the exact committor only when the lag time is a single time step, resulting in a biased estimate in practical applications. We introduce an alternative expression that is minimized by the exact committor at any lag time. The key idea is that, when trajectories enter the stable states, the times that they enter (stopping times) must be used for estimating the committor and transition rate instead of the lag time. Numerical tests on benchmark systems demonstrate that our committor and transition rate estimates are much less sensitive to the choice of lag time. We show how further accuracy for the transition rate can be achieved by combining results from two lag times. We also relate the transition rate expression to a variational approach for kinetic statistics based on the mean-squared residual and discuss further numerical considerations with the aid of a decomposition of the error into dynamic modes.

Understanding the long-time dynamics of complex physical processes depends on our ability to recognize patterns. To simplify the description of these processes, we often introduce a set of reaction coordinates, customarily referred to as collective variables (CVs). The quality of these CVs heavily impacts our comprehension of the dynamics, often influencing the estimates of thermodynamics and kinetics from atomistic simulations. Consequently, identifying CVs poses a fundamental challenge in chemical physics. Recently, significant progress was made by leveraging the predictive ability of unsupervised machine learning techniques to determine CVs. Many of these techniques require temporal information to learn slow CVs that correspond to the long timescale behavior of the studied process. Here, however, we specifically focus on techniques that can identify CVs corresponding to the slowest transitions between states without needing temporal trajectories as input, instead using the spatial characteristics of the data. We discuss the latest developments in this category of techniques and briefly discuss potential directions for thermodynamics-informed spatial learning of slow CVs.

Attention-based neural networks such as transformers have revolutionized various fields such as natural language processing, genomics, and vision. Here, we demonstrate the use of transformers for quantum feedback control through a supervised learning approach. In particular, due to the transformer's ability to capture long-range temporal correlations and training efficiency, we show that it can surpass some of the limitations of previous control approaches, e.g.~those based on recurrent neural networks trained using a similar approach or reinforcement learning. We numerically show, for the example of state stabilization of a two-level system, that our bespoke transformer architecture can achieve unit fidelity to a target state in a short time even in the presence of inefficient measurement and Hamiltonian perturbations that were not included in the training set. We also demonstrate that this approach generalizes well to the control of non-Markovian systems. Our approach can be used for quantum error correction, fast control of quantum states in the presence of colored noise, as well as real-time tuning, and characterization of quantum devices.

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. Phys. Chem. Lett. 2023, 14, 22, 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral map can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

Identifying reaction coordinates(RCs) is an active area of research, given the crucial role RCs play in determining the progress of a chemical reaction. The choice of the reaction coordinate is often based on heuristic knowledge. However, an essential criterion for the choice is that the coordinate should capture both the reactant and product states unequivocally. Also, the coordinate should be the slowest one so that all the other degrees of freedom can easily equilibrate along the reaction coordinate. Also, the coordinate should be the slowest one so that all the other degrees of freedom can easily equilibrate along the reaction coordinate. We used a regularised sparse autoencoder, an energy-based model, to discover a crucial set of reaction coordinates. Along with discovering reaction coordinates, our model also predicts the evolution of a molecular dynamics(MD) trajectory. We showcased that including sparsity enforcing regularisation helps in choosing a small but important set of reaction coordinates. We used two model systems to demonstrate our approach: alanine dipeptide system and proflavine and DNA system, which exhibited intercalation of proflavine into DNA minor groove in an aqueous environment. We model MD trajectory as a multivariate time series, and our latent variable model performs the task of multi-step time series prediction. This idea is inspired by the popular sparse coding approach - to represent each input sample as a linear combination of few elements taken from a set of representative patterns.

Multiscale molecular modeling is widely applied in scientific research of molecular properties over large time and length scales. Two specific challenges are commonly present in multiscale modeling, provided that information between the coarse and fine representations of molecules needs to be properly exchanged: One is to construct coarse grained (CG) models by passing information from the fine to coarse levels; the other is to restore finer molecular details given CG configurations. Although these two problems are commonly addressed independently, in this work, we present a theory connecting them, and develop a methodology called Cycle Coarse Graining (CCG) to solve both problems in a unified manner. In CCG, reconstruction can be achieved via a tractable optimization process, leading to a general method to retrieve fine details from CG simulations, which in turn, delivers a new solution to the CG problem, yielding an efficient way to calculate free energies in a rare-event-free manner. CCG thus provides a systematic way for multiscale molecular modeling, where the finer details of CG simulations can be efficiently retrieved, and the CG models can be improved consistently.

In recent years, the prediction of quantum mechanical observables with machine learning methods has become increasingly popular. Message-passing neural networks (MPNNs) solve this task by constructing atomic representations, from which the properties of interest are predicted. Here, we introduce a method to automatically identify chemical moieties (molecular building blocks) from such representations, enabling a variety of applications beyond property prediction, which otherwise rely on expert knowledge. The required representation can either be provided by a pretrained MPNN, or learned from scratch using only structural information. Beyond the data-driven design of molecular fingerprints, the versatility of our approach is demonstrated by enabling the selection of representative entries in chemical databases, the automatic construction of coarse-grained force fields, as well as the identification of reaction coordinates.

The computer simulation of many molecular processes is complicated by long time scales caused by rare transitions between long-lived states. Here, we propose a new approach to simulate such rare events, which combines transition path sampling with enhanced exploration of configuration space. The method relies on exchange moves between configuration and trajectory space, carried out based on a generalized ensemble. This scheme substantially enhances the efficiency of the transition path sampling simulations, particularly for systems with multiple transition channels, and yields information on thermodynamics, kinetics and reaction coordinates of molecular processes without distorting their dynamics. The method is illustrated using the isomerization of proline in the KPTP tetrapeptide.

Designing manufacturing processes with high yield and strong reliability relies on effective methods for rare event estimation. Genealogical importance splitting reduces the variance of rare event probability estimators by iteratively selecting and replicating realizations that are headed towards a rare event. The replication step is difficult when applied to deterministic systems where the initial conditions of the offspring realizations need to be modified. Typically, a random perturbation is applied to the offspring to differentiate their trajectory from the parent realization. However, this random perturbation strategy may be effective for some systems while failing for others, preventing variance reduction in the probability estimate. This work seeks to address this limitation using a generative model such as a Generative Adversarial Network (GAN) to generate perturbations that are consistent with the attractor of the dynamical system. The proposed GAN-assisted Importance SPlitting method (GANISP) improves the variance reduction for the system targeted. An implementation of the method is available in a companion repository (https://github.com/NREL/GANISP).

Recurrent neural networks have led to breakthroughs in natural language processing and speech recognition. Here we show that recurrent networks, specifically long short-term memory networks can also capture the temporal evolution of chemical/biophysical trajectories. Our character-level language model learns a probabilistic model of 1-dimensional stochastic trajectories generated from higher-dimensional dynamics. The model captures Boltzmann statistics and also reproduces kinetics across a spectrum of timescales. We demonstrate how training the long short-term memory network is equivalent to learning a path entropy, and that its embedding layer, instead of representing contextual meaning of characters, here exhibits a nontrivial connectivity between different metastable states in the underlying physical system. We demonstrate our model’s reliability through different benchmark systems and a force spectroscopy trajectory for multi-state riboswitch. We anticipate that our work represents a stepping stone in the understanding and use of recurrent neural networks for understanding the dynamics of complex stochastic molecular systems. Artificial neural networks have been successfully used for language recognition. Tsai et al. use the same techniques to link between language processing and prediction of molecular trajectories and show capability to predict complex thermodynamics and kinetics arising in chemical or biological physics.