Physics

We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses excursion probability theory to formulate a sequence of Bayesian inverse problems that, when solved, yields the biasing distribution. Solving multiple Bayesian inverse problems can be expensive; more so in higher dimensions. To alleviate the computational cost, we build machine-learning-based surrogates to solve the Bayesian inverse problems that give rise to the biasing distribution. This biasing distribution can then be used in an importance sampling procedure to estimate the extreme excursion probabilities.

We present a simple and effective method for representing periodic functions and enforcing exactly the periodic boundary conditions for solving differential equations with deep neural networks (DNN). The method stems from some simple properties about function compositions involving periodic functions. It essentially composes a DNN-represented arbitrary function with a set of independent periodic functions with adjustable (training) parameters. We distinguish two types of periodic conditions: those imposing the periodicity requirement on the function and all its derivatives (to infinite order), and those imposing periodicity on the function and its derivatives up to a finite order k ( k⩾0 ). The former will be referred to as C ∞ periodic conditions, and the latter C k periodic conditions. We define operations that constitute a C ∞ periodic layer and a C k periodic layer (for any k⩾0 ). A deep neural network with a C ∞ (or C k ) periodic layer incorporated as the second layer automatically and exactly satisfies the C ∞ (or C k ) periodic conditions. We present extensive numerical experiments on ordinary and partial differential equations with C ∞ and C k periodic boundary conditions to verify and demonstrate that the proposed method indeed enforces exactly, to the machine accuracy, the periodicity for the DNN solution and its derivatives.

A new class of multiscale scheme is presented for micro-hydrodynamic problems based on a dual representation of the fluid observables. The hybrid model is first tested against the classical flow between two parallel plates and then applied to a plug flow within a micrometer-sized striction and a shear flow within a microcavity. Both cases demonstrate the capability of the multiscale approach to reproduce the correct macroscopic hydrodynamics also in the presence of refined grids (one and two levels), while retaining the correct thermal fluctuations, embedded in the multiparticle collision method. This provides the first step toward a novel class of fully mesoscale hybrid approaches able to capture the physics of fluids at the micro- and nanoscales whenever a continuum representation of the fluid falls short of providing the complete physical information, due to a lack of resolution and thermal fluctuations.

An efficient computational approach for optimal reconstructing parameters of binary-type physical properties for models in biomedical applications is developed and validated. The methodology includes gradient-based multiscale optimization with multilevel control space reduction by using principal component analysis (PCA) coupled with dynamical control space upscaling. The reduced dimensional controls are used interchangeably at fine and coarse scales to accumulate the optimization progress and mitigate side effects at both scales. Flexibility is achieved through the proposed procedure for calibrating certain parameters to enhance the performance of the optimization algorithm. Reduced size of control spaces supplied with adjoint-based gradients obtained at both scales facilitate the application of this algorithm to models of higher complexity and also to a broad range of problems in biomedical sciences. This technique is shown to outperform regular gradient-based methods applied to fine scale only in terms of both qualities of binary images and computing time. Performance of the complete computational framework is tested in applications to 2D inverse problems of cancer detection by the electrical impedance tomography (EIT). The results demonstrate the efficient performance of the new method and its high potential for minimizing possibilities for false positive screening and improving the overall quality of the EIT-based procedures.

Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this step, a one-dimensional curvilinear stencil is assembled for each edge to perform one-dimensional fast non-uniform WENO interpolation derived in this paper, which is much easier and faster than multi-dimensional interpolation. The second step is to compute the divergence of fluxes at each vertex from the fluxes at nearby edges and vertices by least square approximation of multi-dimensional polynomials. The order of the WENO interpolation in the first step and the degree of the polynomial in the second step determined the order of accuracy of the spatial scheme. After that, explicit RungeKutta time discrete scheme is used to update conservative variables. Several canonical numerical cases were solved to test the accuracy, performance and the capability of shock capturing of the developed method.

Static correlation error(SCE) inevitably emerges when a dissociation of a covalent bond is described with a conventional denstiy-functional theory (DFT) for electrons. SCE gives rise to a serious overshoot in the potential energy at the dissociation limit even in the simplest molecules. The error is attributed to the basic framework of the approximate functional for the exchange correlation energy Exc which refers only to local properties at coordinate r, namely, the electron density n(r) and its derivatives. To solve the problem we developed a functional Ee which uses xc the energy electron distribution ne(e) as a fundamental variable in DFT. ne(e) is obtained by the projection of the density n(r) onto an energy coordinate e defined with the external potential of interest. The functional was applied to the dissociations of single, double, and triple bonds in small molecules showing reasonable agreements with the results given by a high level molecular orbitals theory. We also applied the functional to the computation of the energy change associated with spin depolarization and symmetrization in Carbon atom, which made an improvement over the conventional functional. This work opens the way for development of tougher functional that necessitates non-local properties of electrons such as kinetic energy functional.

The Schrödinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lamé in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac-Weyl equation for a spin-1/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method's convergence in cases where the spectrum in known exactly and apply it to cases where no exact solution exists.

We report on a Python-toolbox for unbiased statistical analysis of fluorescence intermittency properties of single emitters. Intermittency, i.e., step-wise temporal variations in the instantaneous emission intensity and fluorescence decay rate properties are common to organic fluorophores, II-VI quantum dots and perovskite quantum dots alike. Unbiased statistical analysis of intermittency switching time distributions, involved levels and lifetimes is important to avoid interpretation artefacts. This work provides an implementation of Bayesian changepoint analysis and level clustering applicable to time-tagged single-photon detection data of single emitters that can be applied to real experimental data and as tool to verify the ramifications of hypothesized mechanistic intermittency models. We provide a detailed Monte Carlo analysis to illustrate these statistics tools, and to benchmark the extent to which conclusions can be drawn on the photophysics of highly complex systems, such as perovskite quantum dots that switch between a plethora of states instead of just two.

Neutrino experiments study the least understood of the Standard Model particles by observing their direct interactions with matter or searching for ultra-rare signals. The study of neutrinos typically requires overcoming large backgrounds, elusive signals, and small statistics. The introduction of state-of-the-art machine learning tools to solve analysis tasks has made major impacts to these challenges in neutrino experiments across the board. Machine learning algorithms have become an integral tool of neutrino physics, and their development is of great importance to the capabilities of next generation experiments. An understanding of the roadblocks, both human and computational, and the challenges that still exist in the application of these techniques is critical to their proper and beneficial utilization for physics applications. This review presents the current status of machine learning applications for neutrino physics in terms of the challenges and opportunities that are at the intersection between these two fields.

While the particle-in-cell (PIC) method is quite mature, verification and validation of both newly developed methods and individual codes has largely focused on an idiosyncratic choice of a few test cases. Many of these test cases involve either one- or two-dimensional simulations. This is either due to availability of (quasi) analytic solutions or historical reasons. Additionally, tests often focus on investigation of particular physics problems, such as particle emission or collisions, and do not necessarily study the combined impact of the suite of algorithms necessary for a full featured PIC code. As three dimensional (3D) codes become the norm, there is a lack of benchmarks test that can establish the validity of these codes; existing papers either do not delve into the details of the numerical experiment or provide other measurable numeric metrics (such as noise) that are outcomes of the simulation. This paper seeks to provide several test cases that can be used for validation and bench-marking of particle in cell codes in 3D. We focus on examples that are collisionless, and can be run with a reasonable amount of computational power. Four test cases are presented in significant detail; these include, basic particle motion, beam expansion, adiabatic expansion of plasma, and two stream instability. All presented cases are compared either against existing analytical data or other codes. We anticipate that these cases should help fill the void of bench-marking and validation problems and help the development of new particle in cell codes.