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We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification. 

Abstract Biomolecules undergo continuous conformational motions, a subset of which are functionally relevant. Understanding, and ultimately controlling biomolecular function are predicated on the ability to map continuous conformational motions, and identify the functionally relevant conformational trajectories. For equilibrium and near-equilibrium processes, function proceeds along minimum-energy pathways on one or more energy landscapes, because higher-energy conformations are only weakly occupied. With the growing interest in identifying functional trajectories, the need for reliable mapping of energy landscapes has become paramount. In response, various data-analytical tools for determining structural variability are emerging. A key question concerns the veracity with which each data-analytical tool can extract functionally relevant conformational trajectories from a collection of single-particle cryo-EM snapshots. Using synthetic data as an independently known ground truth, we benchmark the ability of four leading algorithms to determine biomolecular energy landscapes and identify the functionally relevant conformational paths on these landscapes. Such benchmarking is essential for systematic progress toward atomic-level movies of continuous biomolecular function. 

A promising new route for structural biology is single-particle imaging with an X-ray Free-Electron Laser (XFEL). This method has the advantage that the samples do not require crystallization and can be examined at room temperature. However, high-resolution structures can only be obtained from a sufficiently large number of diffraction patterns of individual molecules, so-called single particles. Here, we present a method that allows for efficient identification of single particles in very large XFEL datasets, operates at low signal levels, and is tolerant to background. This method uses supervised Geometric Machine Learning (GML) to extract low-dimensional feature vectors from a training dataset, fuse test datasets into the feature space of training datasets, and separate the data into binary distributions of “single particles” and “non-single particles.” As a proof of principle, we tested simulated and experimental datasets of the Coliphage PR772 virus. We created a training dataset and classified three types of test datasets: First, a noise-free simulated test dataset, which gave near perfect separation. Second, simulated test datasets that were modified to reflect different levels of photon counts and background noise. These modified datasets were used to quantify the predictive limits of our approach. Third, an experimental dataset collected at the Stanford Linear Accelerator Center. The single-particle identification for this experimental dataset was compared with previously published results and it was found that GML covers a wide photon-count range, outperforming other single-particle identification methods. Moreover, a major advantage of GML is its ability to retrieve single particles in the presence of structural variability. 

A framework for data assimilation in climate dynamics is presented, combining aspects of quantum mechanics, Koopman operator theory, and kernel methods for machine learning. This approach adapts the formalism of quantum dynamics and measurement to perform data assimilation (filtering), using the Koopman operator governing the evolution of observables as an analog of the Heisenberg operator in quantum mechanics, and a quantum mechanical density operator to represent the data assimilation state. The framework is implemented in a fully empirical, data-driven manner by representing the evolution and measurement operators via matrices in a basis learned from time-ordered observations. Applications to data assimilation of the Nino 3.4 index for the El Nino Southern Oscillation (ENSO) in a comprehensive climate model show promising results.