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Impact of the mesoscale range on error growth and the limits to atmospheric predictability
No full textGlobal numerical weather prediction (NWP) models have begun to resolve the mesoscale k^(-5/3) range of the energy spectrum, which is known to impose an inherently finite range of deterministic predictability per se as errors develop more rapidly on these scales than on the larger scales. However, the dynamics of these errors under the influence of the synoptic-scale k^(-3) range is little studied. Within a perfect-model context, the present work examines the error growth behavior under such a hybrid spectrum in Lorenz’s original model of 1969, and in a series of identical-twin perturbation experiments using an idealized two-dimensional barotropic turbulence model at a range of resolutions. With the typical resolution of today’s global NWP ensembles, error growth remains largely uniform across scales. The theoretically expected fast error growth characteristic of a k^(-5/3) spectrum is seen to be largely suppressed in the first decade of the mesoscale range by the synoptic-scale k^(-3) range. However, it emerges once models become fully able to resolve features on something like a 20-kilometer scale, which corresponds to a grid resolution on the order of a few kilometers
The curious nature of hairpin vortices. Gallery of Fluid Motion.
No full textThe Direct Numerical Simulation (DNS) of an Ekman flow is shown here. It describes the Atmospheric Boundary Layer (ABL) which is the lowest part of the troposphere in direct contact with the Earth's surface.
Applying the vorticity magnitude on xy or xz plane reveals large regions where turbulence is absent in the highly stratified (HS) case (first column of images), a feature which is not present in the neutrally stratified (NS) case (second column of images).
Curiously, if we randomly extract Q-criterion structures from both cases, the hairpins from the HS case appear to be more organized i.e. the head of the hairpin points towards a certain direction.
The final two rows shows several hairpins extracted with Multilevel Percolation (MLP). The first row shows their size measured in terms of ???? (boundary layer height under neutral conditions). The second row shows their position in terms of y+ (viscous wall units).
The Reynolds number is described in terms of the geostrophic wind velocity (G) and ???? as Re = G????/ν
Modeling of chemical reaction systems with detailed balance using gradient structures.
Get PDFWe consider various modeling levels for spatially homogeneous chemical reaction systems,
namely the chemical master equation, the chemical Langevin dynamics, and the reactionrate
equation. Throughout we restrict our study to the case where the microscopic system
satisfies the detailed-balance condition. The latter allows us to enrich the systems with a
gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the
arising links between the associated gradient structures that are driven by the relative entropy
of the detailed-balance steady state. The limit of large volumes is studied in the sense of
evolutionary �-convergence of gradient flows. Moreover, we use the gradient structures to
derive hybrid models for coupling different modeling levels
Machine Learning for Molecular Simulation
No full textMachine learning (ML) is transforming all areas of science. The complex and time-consuming calculations in molecular simulations are particularly suitable for an ML revolution and have already been profoundly affected by the application of existing ML methods. Here we review recent ML methods for molecular simulation, with particular focus on (deep) neural networks for the prediction of quantum-mechanical energies and forces, on coarse-grained molecular dynamics, on the extraction of free energy surfaces and kinetics, and on generative network approaches to sample molecular equilibrium structures and compute thermodynamics. To explain these methods and illustrate open methodological problems, we review some important principles of molecular physics and describe how they can be incorporated into ML structures. Finally, we identify and describe a list of open challenges for the interface between ML and molecular simulation
Free energy computation of particles with membrane-mediated interactions via Langevin dynamics.
Get PDFWe apply well-established concepts of Langevin sampling to derive a new class of algorithms for the efficient computation of free energy differences of fluctuating particles embedded in a ’fast’ membrane, i.e., a membrane that instantaneously adapts to varying particle positions. A geometric poten-
tial accounting for membrane-mediated particle interaction is derived in the framework of variational hybrid models for particles in membranes. Recent explicit representations of the gradient of the geometric interaction poten-
tial allows to apply well-known gradient based Markov Chain Monte-Carlo (MCDC) methods such as Langevin-based sampling
Diffusion maps tailored to arbitrary non-degenerate Ito processes
Get PDFWe present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling
Machine Learning for Molecular Dynamics on Long Timescales
No full textMolecular dynamics (MD) simulation is widely used to analyze the properties of molecules and materials. Most practical applications, such as comparison with experimental measurements, designing drug molecules, or optimizing materials, rely on statistical quantities, which may be prohibitively expensive to compute from direct long-time MD simulations. Classical machine learning (ML) techniques have already had a profound impact on the field, especially for learning low-dimensional models of the long-time dynamics and for devising more efficient sampling schemes for computing long-time statistics. Novel ML methods have the potential to revolutionize long timescale MD and to obtain interpretable models. ML concepts such as statistical estimator theory, end-to-end learning, representation learning, and active learning are highly interesting for the MD researcher and will help to develop new solutions to hard MD problems. With the aim of better connecting the MD and ML research areas and spawning new research on this interface, we define the learning problems in long timescale MD, present successful approaches, and outline some of the unsolved ML problems in this application field
Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds
No full textWe present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches
Large-scale simulation of biomembranes incorporating realistic kinetics into coarse-grained models
No full textBiomembranes are two-dimensional assemblies of phospholipids that are only a few nanometres thick, but form micrometre-sized structures vital to cellular function. Explicit molecular modelling of biologically relevant membrane systems is computationally expensive due to the large number of solvent particles and slow membrane kinetics. Coarse-grained solvent-free membrane models offer efficient sampling but sacrifice realistic kinetics, thereby limiting the ability to predict pathways and mechanisms of membrane processes. Here, we present a framework for integrating coarse-grained membrane models with continuum-based hydrodynamics. This framework facilitates efficient simulation of large biomembrane systems with large timesteps, while achieving realistic equilibrium and non-equilibrium kinetics. It helps to bridge between the nanometer/nanosecond spatiotemporal resolutions of coarse-grained models and biologically relevant time- and lengthscales. As a demonstration, we investigate fluctuations of red blood cells, with varying cytoplasmic viscosities, in 150-milliseconds-long trajectories, and compare kinetic properties against single-cell experimental observations
Non- markovian data-driven modeling of single-cell motility
No full textTrajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process