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Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach
No full textABSTRACT
We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow Chen et al. [J. Math. Phys. 56, 113302 (2015)] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower (“target”) or the higher (“simulation”) temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observations from the perspective of stochastic optimization algorithms and enhanced sampling schemes in molecular dynamics
Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry
No full textWe derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications
A General Framework for Machine Learning based Optimization Under Uncertainty
No full textWe propose a general framework for machine learning based optimization under uncertainty. Our approach replaces the complex forward model by a surrogate, e.g., a neural network, which is learned simultaneously in a one-shot sense when solving the optimal control problem. Our approach relies on a reformulation of the problem as a penalized empirical risk minimization problem for which we provide a consistency analysis in terms of large data and increasing penalty parameter. To solve the resulting problem, we suggest a stochastic gradient method with adaptive control of the penalty parameter and prove convergence under suitable assumptions on the surrogate model. Numerical experiments illustrate the results for linear and nonlinear surrogate models
A Quasi-Monte Carlo Method for Optimal Control Under Uncertainty
No full textWe study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are nonconvex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error, and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments
Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective
Get PDFMacroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative
systems or gradient flows) have a natural variational structure, which can be derived from the
large-deviation rate functional for the density of the particle system. While large deviations can be
studied in considerable generality, these variational structures are often restricted to systems in detailed
balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this
variational connection beyond dissipative systems by augmenting densities with fluxes, which encode
non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density
cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a
generalised orthogonality relation between them. We then apply this abstract theory to various stochastic
particle systems – independent copies of jump processes, zero-range processes, chemical-reaction networks
in complex balance and lattice-gas models
Quantum dynamics of a polar rotor acted upon by an electric rectangular pulse of variable duration
No full textAs demonstrated in our previous work [J. Chem. Phys. 149, 174109 (2018)], the kinetic energy imparted to a quantum rotor by a non-resonant electromagnetic pulse with a Gaussian temporal profile exhibits quasi-periodic drops as a function of the pulse duration. Herein, we show that this behavior can be reproduced with a simple waveform, namely a rectangular electric pulse of variable duration, and examine, both numerically and analytically, its causes. Our analysis reveals that the drops result from the oscillating populations that make up the wavepacket created by the pulse and that they are necessarily accompanied by drops in the orientation and by a restoration of the pre-pulse alignment of the rotor. Handy analytic formulae are derived that allow to predict the pulse durations leading to diminished kinetic energy transfer and orientation. Experimental scenarios are discussed where the phenomenon could be utilized or be detrimental
Hydrodynamic coupling for particle-based solvent-free membrane models
No full textThe great challenge with biological membrane systems is the wide range of scales involved, from nanometers and picoseconds for individual lipids to the micrometers and beyond millisecond for cellular signaling processes. While solvent-free coarse-grained membrane models are convenient for large-scale simulations and promising to provide insight into slow processes involving membranes, these models usually have unrealistic kinetics. One major obstacle is the lack of an equally convenient way of introducing hydrodynamic coupling without significantly increasing the computational cost of the model. To address this, we introduce a framework based on anisotropic Langevin dynamics, for which major in-plane and out-of-plane hydrodynamic effects are modeled via friction and diffusion tensors from analytical or semi-analytical solutions to Stokes hydrodynamic equations. Using this framework, in conjunction with our recently developed membrane model, we obtain accurate dispersion relations for planar membrane patches, both free-standing and in the vicinity of a wall. We briefly discuss how non-equilibrium dynamics is affected by hydrodynamic interactions. We also measure the surface viscosity of the model membrane and discuss the affecting dissipative mechanisms
Genome-wide Determination Of Splicing Efficiency And Dynamics From RNA-Seq Data
No full textEukaryotic genes are mostly composed of a series of exons intercalated by sequences with no coding potential called introns. These sequences are generally removed from primary transcripts to form mature RNA molecules in a post-transcriptional process called splicing. An efficient splicing of primary transcripts is an essential step in gene expression and its misregulation is related to numerous human diseases. Thus, to better understand the dynamics of this process and the perturbations that might be caused by aberrant transcript processing, it is important to quantify splicing efficiency. In this thesis, I introduce SPLICE-q, a fast and user-friendly Python tool for genome-wide SPLICing Efficiency quantification. It supports studies focusing on the implications of splicing efficiency in transcript processing dynamics. SPLICE-q uses aligned reads from RNA-Seq to quantify splicing efficiency for each intron individually and allows the user to select different levels of restrictiveness concerning the introns’ overlap with other genomic elements, such as exons from other genes. I demonstrate SPLICE-q’s application using three use cases including two different species and methodologies. These analyses illustrate that SPLICE-q can detect a progressive increase of splicing efficiency throughout a time course of nascent RNA-Seq and it might be useful when it comes to understanding cancer progression beyond mere gene expression levels. Furthermore, I provide an in-depth study of time course nascent BrU-Seq data to address questions concerning differences in the speed of splicing and the underlying biological features that might be associated with it. SPLICE-q and its documentation are publicly available at: https://github.com/vrmelo/SPLICE-q
Markov models from the square root approximation of the Fokker–Planck equation: calculating the grid-dependent flux
Get PDFAbstract
Molecular dynamics (MD) are extremely complex, yet understanding the slow components of
their dynamics is essential to understanding their macroscopic properties. To achieve this, one
models the MD as a stochastic process and analyses the dominant eigenfunctions of the
associated Fokker–Planck operator, or of closely related transfer operators. So far, the
calculation of the discretized operators requires extensive MD simulations. The square-root
approximation of the Fokker–Planck equation is a method to calculate transition rates as a
ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be
calculated without a simulation. In a previous work we still used MD simulations to determine
the flux. Here, we propose several methods to calculate the exact or approximate flux for
various grid types, and thus estimate the rate matrix without a simulation. Using model
potentials we test computational efficiency of the methods, and the accuracy with which they
reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate
matrices with up to O(106) states can be obtained within seconds on a single
high-performance compute server if regular grids are used
NUMERICAL SIMULATION OF MULTISCALE FAULT SYSTEMS WITH RATE- AND STATE-DEPENDENT FRICTION
Get PDFAbstract. We consider the deformation of a geological structure with non-intersecting
faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and
state-depending friction conditions along the common interfaces. We derive a mathematical
model that contains classical Dieterich- and Ruina-type friction as special cases and accounts
for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme
leads to a coupled system of non-smooth, convex minimization problems for rate and state
to be solved in each time step. Additional spatial discretization by a mortar method and
piecewise constant finite elements allows for the decoupling of rate and state by a fixed
point iteration and efficient algebraic solution of the rate problem by truncated non-smooth
Newton methods. Numerical experiments with a spring slider and a layered multiscale system
illustrate the behavior of our model as well as the efficiency and reliability of the numerical
solver