Let $P$ be a linear differential operator over $\mathcal{D} \subset \mathbb{R}^d$ and $U = (U_x)_{x \in \mathcal{D}}$ a second order stochastic process. In the first part of this article, we prove a new simple necessary and sufficient condition for all the trajectories of $U$ to verify the partial differential equation (PDE) $T(U) = 0$. This condition is formulated in terms of the covariance kernel of $U$. The novelty of this result is that the equality $T(U) = 0$ is understood in the sense of distributions, which is a functional analysis framework particularly adapted to the study of PDEs. This theorem provides precious insights during the second part of this article, which is dedicated to performing "physically informed" machine learning on data that is solution to the homogeneous 3 dimensional free space wave equation. We perform Gaussian Process Regression (GPR) on this data, which is a kernel based Bayesian approach to machine learning. To do so, we put Gaussian process (GP) priors over the wave equation's initial conditions and propagate them through the wave equation. We obtain explicit formulas for the covariance kernel of the corresponding stochastic process; this kernel can then be used for GPR. We explore two particular cases : the radial symmetry and the point source. For the former, we derive convolution-free GPR formulas; for the latter, we show a direct link between GPR and the classical triangulation method for point source localization used e.g. in GPS systems. Additionally, this Bayesian framework gives rise to a new answer for the ill-posed inverse problem of reconstructing initial conditions for the wave equation with finite dimensional data, and simultaneously provides a way of estimating physical parameters from this data as in [Raissi et al,2017]. We finish by showcasing this physically informed GPR on a number of practical examples.
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Processing 是一門開(kai)源(yuan)編程語言和與(yu)之配套的集成開(kai)發環境（IDE）的名稱。Processing 在電子藝術和視覺設(she)計(ji)社區被(bei)用(yong)來(lai)教授編程基礎，并運用(yong)于(yu)大量的新媒體(ti)和互動藝術作品中。
Robust estimation of a mean vector, a topic regarded as obsolete in the traditional robust statistics community, has recently surged in machine learning literature in the last decade. The latest focus is on the sub-Gaussian performance and computability of the estimators in a non-asymptotic setting. Numerous traditional robust estimators are computationally intractable, which partly contributes to the renewal of the interest in the robust mean estimation. Robust centrality estimators, however, include the trimmed mean and the sample median. The latter has the best robustness but suffers a low-efficiency drawback. Trimmed mean and median of means, %as robust alternatives to the sample mean, and achieving sub-Gaussian performance have been proposed and studied in the literature. This article investigates the robustness of leading sub-Gaussian estimators of mean and reveals that none of them can resist greater than $25\%$ contamination in data and consequently introduces an outlyingness induced winsorized mean which has the best possible robustness (can resist up to $50\%$ contamination without breakdown) meanwhile achieving high efficiency. Furthermore, it has a sub-Gaussian performance for uncontaminated samples and a bounded estimation error for contaminated samples at a given confidence level in a finite sample setting. It can be computed in linear time.
We provide guarantees for approximate Gaussian Process (GP) regression resulting from two common low-rank kernel approximations: based on random Fourier features, and based on truncating the kernel's Mercer expansion. In particular, we bound the Kullback-Leibler divergence between an exact GP and one resulting from one of the afore-described low-rank approximations to its kernel, as well as between their corresponding predictive densities, and we also bound the error between predictive mean vectors and between predictive covariance matrices computed using the exact versus using the approximate GP. We provide experiments on both simulated data and standard benchmarks to evaluate the effectiveness of our theoretical bounds.
We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [Numer. Math. 144, 451--477, 2020] to the convection-dominated regime. Slightly adjusting the stabilized finite element method proposed for dominant diffusion, we draw upon a local error analysis to obtain quasi-optimal convergence along the characteristics of the convective field through the data set. The weight function multiplying the discrete solution is taken to be Lipschitz and a corresponding super approximation result (discrete commutator property) is proven. The effect of data perturbations is included in the analysis and we conclude the paper with some numerical experiments.
We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.
This paper introduces a local-to-unity/small sigma process for a stationary time series with strong persistence and non-negligible long run risk. This process represents the stationary long run component in an unobserved short- and long-run components model involving different time scales. More specifically, the short run component evolves in the calendar time and the long run component evolves in an ultra long time scale. We develop the methods of estimation and long run prediction for the univariate and multivariate Structural VAR (SVAR) models with unobserved components and reveal the impossibility to consistently estimate some of the long run parameters. The approach is illustrated by a Monte-Carlo study and an application to macroeconomic data.
We consider the problem of estimating the autocorrelation operator of an autoregressive Hilbertian process. By means of a Tikhonov approach, we establish a general result that yields the convergence rate of the estimated autocorrelation operator as a function of the rate of convergence of the estimated lag zero and lag one autocovariance operators. The result is general in that it can accommodate any consistent estimators of the lagged autocovariances. Consequently it can be applied to processes under any mode of observation: complete, discrete, sparse, and/or with measurement errors. An appealing feature is that the result does not require delicate spectral decay assumptions on the autocovariances but instead rests on natural source conditions. The result is illustrated by application to important special cases.
In this paper, we consider the motion energy minimization problem for a robot that uses millimeter-wave (mm-wave) communications assisted by an intelligent reflective surface (IRS). The robot must perform tasks within given deadlines and it is subject to uplink quality of service (QoS) constraints. This problem is crucial for fully automated factories that are governed by the binomial of autonomous robots and new generations of mobile communications, i.e., 5G and 6G. In this new context, robot energy efficiency and communication reliability remain fundamental problems that couple in optimizing robot trajectory and communication QoS. More precisely, to account for the mutual dependency between robot position and communication QoS, robot trajectory and beamforming at the IRS and access point all need to be optimized. We present a solution that can decouple the two problems by exploiting mm-wave channel characteristics. Then, a closed-form solution is obtained for the beamforming optimization problem, whereas the trajectory is optimized by a novel successive-convex optimization-based algorithm that can deal with abrupt line-of-sight (LOS) to non-line-of-sight (NLOS) transitions. Specifically, the algorithm uses a radio map to avoid collisions with obstacles and poorly covered areas. We prove that the algorithm can converge to a solution satisfying the Karush-Kuhn-Tucker conditions. The simulation results show a fast convergence rate of the algorithm and a dramatic reduction of the motion energy consumption with respect to methods that aim to find maximum-rate trajectories. Moreover, we show that the use of passive IRSs represents a powerful solution to improve the radio coverage and motion energy efficiency of robots.
This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal H\"older continuity of the spatial semi-discrete numerical solution.
An abundant amount of data gathered during wind tunnel testing and health monitoring of structures inspires the use of machine learning methods to replicate the wind forces. This paper presents a data-driven Gaussian Process-Nonlinear Finite Impulse Response (GP-NFIR) model of the nonlinear self-excited forces acting on structures. Constructed in a nondimensional form, the model takes the effective wind angle of attack as lagged exogenous input and outputs a probability distribution of the forces. The nonlinear input/output function is modeled by a GP regression. Consequently, the model is nonparametric, thereby circumventing to set up the function's structure a priori. The training input is designed as random harmonic motion consisting of vertical and rotational displacements. Once trained, the model can predict the aerodynamic forces for both prescribed input motion and aeroelastic analysis. The concept is first verified for a flat plate's analytical solution by predicting the self-excited forces and flutter velocity. Finally, the framework is applied to a streamlined and bluff bridge deck based on Computational Fluid Dynamics (CFD) data. The model's ability to predict nonlinear aerodynamic forces, flutter velocity, and post-flutter behavior are highlighted. Applications of the framework are foreseen in the structural analysis during the design and monitoring of slender line-like structures.
We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms of the local geometry of the given constraint set $K$. Our main result shows that the minimax risk (up to constant factors) under the squared $L_2$ loss is given by $\epsilon^{*2} \wedge \operatorname{diam}(K)^2$ with \begin{align*} \epsilon^* = \sup \bigg\{\epsilon : \frac{\epsilon^2}{\sigma^2} \leq \log M^{\operatorname{loc}}(\epsilon)\bigg\}, \end{align*} where $\log M^{\operatorname{loc}}(\epsilon)$ denotes the local entropy of the set $K$, and $\sigma^2$ is the variance of the noise. We utilize our abstract result to re-derive known minimax rates for some special sets $K$ such as hyperrectangles, ellipses, and more generally quadratically convex orthosymmetric sets. Finally, we extend our results to the unbounded case with known $\sigma^2$ to show that the minimax rate in that case is $\epsilon^{*2}$.
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