18 research outputs found
Combinatorial Mori-Zwanzig Theory
We introduce a combinatorial version Mori-Zwanzig theory and develop from it
a family of self-consistent evolution equations for the correlation function or
Green's function of interactive many-body systems. The core idea is to use an
ansatz to rewrite the memory kernel (self-energy) of the regular Mori-Zwanzig
equation as a function composition of the correlation (Green's) function. Then
a series of algebraic combinatorial tools, especially the commutative and
noncommutative Bell polynomials, are used to determine the exact Taylor series
expansion of the composition function. The resulting combinatorial Mori-Zwanzig
equation (CMZE) yields novel non-perturbative expansions of the equation of
motion for the correlation (Green's) function. The structural equation for
deriving such a combinatorial expansion resembles the combinatorial
Dyson-Schwinger equation and may be viewed as its temporal-domain analogue.
After introducing the abstract word and tree representation of the CMZE, we
show its wide-range application in classical, stochastic, and quantum many-body
systems. In all these examples, the new self-consistent expansions we obtained
with the CMZE are similar to the diagrammatic skeleton expansions used in
quantum many-body theory and lattice statistical field theory. We expect such a
new framework can be used to calculate the correlation (Green's) function for
strongly correlated/interactive many-body systems
Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations
We develop a thorough mathematical analysis of the effective Mori-Zwanzig
(EMZ) equation governing the dynamics of noise-averaged observables in
stochastic differential equations driven by multiplicative Gaussian white
noise. Building upon recent work on hypoelliptic operators, we prove that the
EMZ memory kernel and fluctuation terms converge exponentially fast in time to
a unique equilibrium state which admits an explicit representation. We apply
the new theoretical results to the Langevin dynamics of a high-dimensional
particle system with smooth interaction potential.Comment: 22 pages, 1 figur
Learning Stochastic Dynamics with Statistics-Informed Neural Network
We introduce a machine-learning framework named statistics-informed neural
network (SINN) for learning stochastic dynamics from data. This new
architecture was theoretically inspired by a universal approximation theorem
for stochastic systems, which we introduce in this paper, and the
projection-operator formalism for stochastic modeling. We devise mechanisms for
training the neural network model to reproduce the correct \emph{statistical}
behavior of a target stochastic process. Numerical simulation results
demonstrate that a well-trained SINN can reliably approximate both Markovian
and non-Markovian stochastic dynamics. We demonstrate the applicability of SINN
to coarse-graining problems and the modeling of transition dynamics.
Furthermore, we show that the obtained reduced-order model can be trained on
temporally coarse-grained data and hence is well suited for rare-event
simulations
Detecting Label Noise via Leave-One-Out Cross-Validation
We present a simple algorithm for identifying and correcting real-valued
noisy labels from a mixture of clean and corrupted sample points using Gaussian
process regression. A heteroscedastic noise model is employed, in which
additive Gaussian noise terms with independent variances are associated with
each and all of the observed labels. Optimizing the noise model using maximum
likelihood estimation leads to the containment of the GPR model's predictive
error by the posterior standard deviation in leave-one-out cross-validation. A
multiplicative update scheme is proposed for solving the maximum likelihood
estimation problem under non-negative constraints. While we provide proof of
convergence for certain special cases, the multiplicative scheme has
empirically demonstrated monotonic convergence behavior in virtually all our
numerical experiments. We show that the presented method can pinpoint corrupted
sample points and lead to better regression models when trained on synthetic
and real-world scientific data sets