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Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…

We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…

Interval analysis (or interval bound propagation, IBP) is a popular technique for verifying and training provably robust deep neural networks, a fundamental challenge in the area of reliable machine learning. However, despite substantial…

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential…

Our focus is on robust recovery algorithms in statistical linear inverse problem. We consider two recovery routines - the much studied linear estimate originating from Kuks and Olman [42] and polyhedral estimate introduced in [37]. It was…

We survey old and new conjectures and results on various types of spherical maximal functions, emphasizing problems with a fractal dilation set.

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner.…

Spin glasses are paradigmatic models that deliver concepts relevant for a variety of systems. However, rigorous analytical results are difficult to obtain for spin-glass models, in particular for realistic short-range models. Therefore…

We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand the smoothness of what we call the spectral density function…

This review presents various aspects of a mean-field spin glass model known as the p-spin spherical spin glass model, which has raised a lot of interest in the study of spin glasses, and also for its possible links with a mean-field theory…

We present a simplified and unified analysis of the Johnson-Lindenstrauss (JL) lemma, a cornerstone of dimensionality reduction for managing high-dimensional data. Our approach simplifies understanding and unifies various constructions…

We address a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solution is sought in the function space spanned by physics-informed random projection neural…

The work deals with the existence of solutions of a certain system of quadratic integral equations in H^2(R^d,R^N), d = 2, 3. We demonstrate the existence of a perturbed solution by virtue of a fixed point technique.

Differential analysis aims at inferring global properties of nonlinear behaviors from the local analysis of the linearized dynamics. The paper motivates and illustrates the use of differential analysis on the nonlinear pendulum model, an…

Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore,…

We consider a non-polynomial cubic spline to develop the classes of methods for the numerical solution of singularly perturbed two-point boundary value problems. The proposed methods are second and fourth order accurate and applicable to…

Parameter estimation for differential equations from measured data is an inverse problem prevalent across quantitative sciences. Physics-Informed Neural Networks (PINNs) have emerged as effective tools for solving such problems, especially…

The goal of this paper is to provide a simple approach to perform local sensitivity analysis using Physics-informed neural networks (PINN). The main idea lies in adding a new term in the loss function that regularizes the solution in a…

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in…

The paper is focused on the nonlinear stability analysis of stochastic $\theta$-methods. In particular, we consider nonlinear stochastic differential equations such that the mean-square deviation between two solutions exponentially decays,…