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The Fokker--Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines, but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants and hence require the development of a data-driven modeling approach. When the data available is directly on the PDF, there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physicsinformed neural networks (PINNs) that introduces a new loss function using the Kullback--Leibler divergence to connect the stochastic samples with the FP equation to simultaneously learn the equation and infer the multidimensional PDF at all times. In particular, we consider two types of inverse problems, type I, where the FP equation is known but the initial PDF is unknown, and type II, in which, in addition to the unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or L\'evy noise or a combination of both. We demonstrate the new PINN framework in detail in the one-dimensional (1D) case, but we also provide results for up to five dimensions demonstrating that we can infer both the FP equation and dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles. [ABSTRACT FROM AUTHOR]

We present a computational stochastic methodology for generating and optimizing random meta-material (MM) configurations with nonoverlapping geometric constraints subject to various types of covariances and distributions characterizing the randomness of the MM configura- tions. The methodology developed consists of three main components: (1) a deterministic solver for electromagnetic scattering from multiple objects, (2) the Karhunen-Loève (K-L) expansion to represent the correlated configurations of the scattering objects, and (3) the multi-element proba- bilistic collocation method (ME-PCM) to provide flexibility in achieving desired distributions of the MMs. In the current work, we employ random fields from a Spartan family that includes covariance functions with damped oscillatory behavior. The algorithm is applied to study light propagation through random layered heterojunctions and random 3-D MMs. We found that greater transmission and reflection, compared to the uniformly spaced structures, can be achieved for a structure with an oscillatory spacing profile along the propagation direction. Optimized configurations of the hetero- junctions and 3-D MMs have been found with larger or smaller transmission coefficients for different wave numbers of the incoming wave and different correlations. [ABSTRACT FROM AUTHOR]

We introduce the concept of numerical Gaussian processes, which we define as Gauss- ian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where (a) all we observe are noisy data on black-box initial conditions, and (b) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time- dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time- dependent operators. In all examples, we are able to recover accurate approximations of the latent solutions, and consistently propagate uncertainty, even in cases involving very long time integration. [ABSTRACT FROM AUTHOR]

We develop a framework for multifidelity information fusion and predictive inference in high-dimensional input spaces and in the presence of massive data sets. Hence, we tackle simultaneously the "big N" problem for big data and the curse of dimensionality in multivariate parametric problems. The proposed methodology establishes a new paradigm for constructing response surfaces of high-dimensional stochastic dynamical systems, simultaneously accounting for multifidelity in physical models as well as multifidelity in probability space. Scaling to high dimensions is achieved by data-driven dimensionality reduction techniques based on hierarchical functional decompositions and a graph-theoretic approach for encoding custom autocorrelation structure in Gaussian process priors. Multifidelity information fusion is facilitated through stochastic autoregressive schemes and frequency-domain machine learning algorithms that scale linearly with the data. Taking together these new developments leads to linear complexity algorithms as demonstrated in benchmark problems involving deterministic and stochastic fields in up to 105 input dimensions and 105 training points on a standard desktop computer. [ABSTRACT FROM AUTHOR]

We generalize existing Jacobi-Gauss-Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μPa,b j (x), where μ > -1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals and derivatives of these weighted Jacobi polynomials, hence extending the three-term recurrence relationship of Jacobi polynomials. The new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities. We demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter μ, even for cases where we do not know explicitly the form of singularity in the solution at the boundaries. [ABSTRACT FROM AUTHOR]

We propose three time-splitting schemes for nonlinear time-fractional differential equations with smooth solutions, where the order of the fractional derivative is 0 < α < 1. While one of the schemes is of order α, the other two schemes are of order 1 + α and 2 - α and thus they can be combined to provide flexible numerical methods with convergence order no less than 3/2. We prove the convergence and stability of the proposed schemes. Numerical examples illustrate the flexibility and the efficiency of these time-splitting schemes and show that they work for multirate and stiff time-fractional differential systems effectively. [ABSTRACT FROM AUTHOR]

Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Lévy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable L´evy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm-Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order v ∈ (0, 2). In the regular case, the corresponding tempered differential operators are associated with tempering functions pI (x) = exp(2τ) and pII (x) = exp(-2τ), τ ≥ 0, respectively, in the regular TFSLP-I and TFSLP-II, which do not vanish in [-1, 1]. In contrast, the corresponding differential operators in the singular setting are associated with different forms of pI (x) = exp(2τ)(1-x)1+α(1+ x)1+β and pII (x) = exp(-2τ)(1 - x)1+α(1 + x)1+β, vanishing at x = ±1 in the singular TFSLP-I and TFSLP-II, respectively. The aforementioned tempered fractional differential operators are both of tempered Riemann--Liouville and tempered Caputo type of fractional order μ = v/2 ∈ (0, 1). We prove the well-posedness of the boundary-value problems and that the eigenvalues of the regular tempered problems are real-valued and the corresponding eigenfunctions are orthogonal. Next, we obtain the explicit eigensolutions to TFSLP-I and -II as nonpolynomial functions, which we define as tempered Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with TFSLP-I and -II. Finally, we introduce these eigenfunctions as new basis (and test) functions for spectrally accurate approximation of functions and tempered-fractional differential operators. To this end, we further develop a Petrov-Galerkin spectral method for solving tempered fractional ODEs, followed by the corresponding stability and convergence analysis, which validates the achieved spectral convergence in our simulations. [ABSTRACT FROM AUTHOR]