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A field equation for induction-transduction of activation-deactivation probability on measurable space

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arXiv 2022年

作者： Bastian, Caleb Deen Rabitz, Herschel Program in Applied and Computational Mathematics Princeton University Princeton NJ08544 United States Department of Chemistry Princeton University Princeton NJ08544 United States

Induction-transduction of activating-deactivating points are fundamental mechanisms of action that underlie innumerable systems and phenomena, mathematical, natural, and anthropogenic, and can exhibit complex behaviors such as self-excitation, phase transitions, hysteresis, polarization, periodicity, chaos, wave behavior, geometry, and energy transfer. We describe a class of primitives for induction-transduction based on dynamics on images of marked random counting measures under graphical random transformations. We derive a field equation for the law of the activation-deactivation (Bernoulli) process on an arbitrary measurable space and describe some mechanisms of action on the unit *** Codes 37A50, 60G57, 60G60, 60G55 © 2022, CC BY-NC-SA.

关键词： Energy transfer

Unbiased Estimation of the Vanilla and Deterministic Ensemble Kalman–Bucy Filters

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arXiv 2022年

作者： Álvarez Ballesteros, Miguel Ángel Chada, Neil K. Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia

In this article we consider the development of an unbiased estimator for the ensemble Kalman–Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology which can be viewed as a continuous-time analogue of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work [Rhee & Glynn 2010, [31]] which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization, and through the number of samples at each level. Our estimator will be specific to linear and Gaussian settings, where we know that the EnKBF is consistent, in the particle limit N → ∞, with the KBF. We highlight this for two particular variants of the EnKBF, i.e. the deterministic and vanilla variants, and demonstrate this on a linear Ornstein–Uhlenbeck process. We compare this with the EnKBF and the multilevel (MLEnKBF), for experiments with varying dimension size. We also provide a proof of the multilevel deterministic EnKBF, which provides a guideline for some of the unbiased methods. © 2022, CC BY-NC-SA.

关键词： Continuous time systems

random graphs by product random measures

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arXiv 2022年

作者： Bastian, Caleb Deen Rabitz, Herschel Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States

A natural representation of random graphs is the random measure. The collection of product random measures, their transformations, and non-negative test functions forms a general representation of the collection of non-negative weighted random graphs, directed or undirected, labeled or unlabeled, where (i) the composition of the test function and transformation is a non-negative edge weight function, (ii) the mean measures encode edge density/weight and vertex degree density/weight, and (iii) the mean edge weight, when square-integrable, encodes generalized spectral and Sobol representations. We develop a number of properties of these random graphs, and we give simple examples of some of their possible *** Codes 60G57, 60G55 © 2022, CC BY-NC-SA.

关键词： Encoding (symbols)

ON THE SOLUTION OF LAPLACE’S EQUATION IN THE VICINITY OF TRIPLE JUNCTIONS

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Pure and applied Analysis 2020年第2期2卷 447-476页

作者： Hoskins, Jeremy Rachh, Manas Applied Mathematics Program Yale University New Haven CT United States Center for Computational Mathematics Flatiron Institute New York NY United States

We characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider triple junctions, i.e., points at which three distinct media meet. We show that, under suitable conditions, solutions to the boundary integral equations in the vicinity of a triple junction are well-approximated by linear combinations of functions of the form t^β, where t is the distance of the point from the junction and the powers β depend only on the material properties of the media and the angles at which their boundaries meet. Moreover, we use this analysis to design efficient discretizations of boundary integral equations for Laplace transmission problems in regions with triple junctions and demonstrate the accuracy and efficiency of this algorithm with a number of examples. © 2020, Mathematical Sciences Publishers. All rights reserved.

关键词： boundary integral equations corners multiple junction interfaces potential theory singular solutions

Modeling High-Resolution Spatio-Temporal Wind with Deep Echo State Networks and Stochastic Partial Differential Equations

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arXiv 2024年

作者： Wang, Kesen Kim, Minwoo Castruccio, Stefano Genton, Marc G. Statistics Program King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistics Pusan National University Busan46241 Korea Republic of Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

In the past decades, clean and renewable energy has gained increasing attention due to a global effort on carbon footprint reduction. In particular, Saudi Arabia is gradually shifting its energy portfolio from an exclusive use of oil to a reliance on renewable energy, and, in particular, wind. Modeling wind for assessing potential energy output in a country as large, geographically diverse and understudied as Saudi Arabia is a challenge which implies highly non-linear dynamic structures in both space and time. To address this, we propose a spatio-temporal model whose spatial information is first reduced via an energy distance-based approach and then its dynamical behavior is informed by a sparse and stochastic recurrent neural network (Echo State Network). Finally, the full spatial data is reconstructed by means of a non-stationary stochastic partial differential equation-based approach. Our model can capture the fine scale wind structure and produce more accurate forecasts of both wind speed and energy in lead times of interest for energy grid management and save annually as much as one million dollar against the closest competitive model. © 2024, CC BY.

关键词： Clean energy

The phase diagram of a deep potential water model

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arXiv 2021年

作者： Zhang, Linfeng Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

Using the Deep Potential methodology, we construct a model that reproduces accurately the potential energy surface of the SCAN approximation of density functional theory for water, from low temperature and pressure to about 2400 K and 50 GPa, excluding the vapor stability region. The computational efficiency of the model makes it possible to predict its phase diagram using molecular dynamics. Satisfactory overall agreement with experimental results is obtained. The fluid phases, molecular and ionic, and all the stable ice polymorphs, ordered and disordered, are predicted correctly, with the exception of ice III and XV that are stable in experiments, but metastable in the model. The evolution of the atomic dynamics upon heating, as ice VII transforms first into ice VII′′ and then into an ionic fluid, reveals that molecular dissociation and breaking of the ice rules coexist with strong covalent fluctuations, explaining why only partial ionization was inferred in experiments. © 2021, CC BY-NC-ND.

关键词： Phase diagrams

Ab Initio Generalized Langevin Equation

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arXiv 2022年

作者： Xie, Pinchen Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States AI for Science Institute Beijing China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing China

We introduce a machine learning-based approach called ab initio generalized Langevin equation (AIGLE) to model the dynamics of slow collective variables in materials and molecules. In this scheme, the parameters are learned from atomistic simulations based on ab initio quantum mechanical models. Force field, memory kernel, and noise generator are constructed in the context of the Mori-Zwanzig formalism, under the constraint of the fluctuation-dissipation theorem. Combined with deep potential molecular dynamics and electronic density functional theory, this approach opens the way to multi-scale modeling in a variety of situations. Here, we demonstrate this capability with a study of two mesoscale processes in crystalline lead titanate, namely the field-driven dynamics of a planar ferroelectric domain wall, and the dynamics of an extensive lattice of coarse-grained electric dipoles. In the first case, AIGLE extends the reach of ab initio simulations to a regime of noise-driven motions not accessible to molecular dynamics. In the second case, AIGLE deals with an extensive set of collective variables by adopting a local approximation for the memory kernel and retaining only short-range noise correlations. The scheme is computationally more efficient than molecular dynamics by several orders of magnitude, and mimics the microscopic dynamics at low frequencies where it reproduces accurately the dominant far-infrared absorption frequency. © 2022, CC BY.

关键词： Molecular dynamics

Unbiased Estimation using Underdamped Langevin Dynamics

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arXiv 2022年

作者： Ruzayqat, Hamza Chada, Neil K. Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work we consider the unbiased estimation of expectations w.r.t. probability measures that have non-negative Lebesgue density, and which are known point-wise up-to a normalizing constant. We focus upon developing an unbiased method via the underdamped Langevin dynamics, which has proven to be popular of late due to applications in statistics and machine learning. Specifically in continuous-time, the dynamics can be constructed so that as the time goes to infinity they admit the probability of interest as a stationary measure. In many cases, time-discretized versions of the underdamped Langevin dynamics are used in practice which are run only with a fixed number of iterations. We develop a novel scheme based upon doubly randomized estimation as in [17, 19], which requires access only to time-discretized versions of the dynamics. The proposed scheme aims to remove the dicretization bias and the bias resulting from running the dynamics for a finite number of iterations. We prove, under standard assumptions, that our estimator is of finite variance and either has finite expected cost, or has finite cost with a high probability. To illustrate our theoretical findings we provide numerical experiments which verify our theory, which include challenging examples from Bayesian statistics and statistical *** Codes 60J22, 65C05, 65C40, 82C31, 62G08, 35Q56 © 2022, CC BY.

关键词： Dynamics

Existence and Stability of Nonequilibrium Steady States of Nernst-Planck-Navier-Stokes Systems

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arXiv 2022年

作者： Constantin, Peter Ignatova, Mihaela Lee, Fizay-Noah Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Temple University PhiladelphiaPA19122 United States Program In Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

MSC Codes 35Q30, 35Q35, 35Q92We consider the Nernst-Planck-Navier-Stokes system in a bounded domain of Rd, d = 2, 3 with general nonequilibrium Dirichlet boundary conditions for the ionic concentrations. We prove the existence of smooth steady state solutions and present a sufficient condition in terms of only the boundary data that guarantees that these solutions have nonzero fluid velocity. We show that time evolving solutions are ultimately bounded uniformly, independently of their initial size. In addition, we consider one dimensional steady states with steady nonzero currents and show that they are globally nonlinearly stable as solutions in a three dimensional periodic strip, if the currents are sufficiently weak. © 2022, CC BY.

关键词： Navier Stokes equations