Related papers: Benign overfitting in Fixed Dimension via Physics-…

The widely observed 'benign overfitting phenomenon' in the neural network literature raises the challenge to the 'bias-variance trade-off' doctrine in the statistical learning theory. Since the generalization ability of the 'lazy trained'…

We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies…

As a typical application of deep learning, physics-informed neural network (PINN) {has been} successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great…

In many modern applications of deep learning the neural network has many more parameters than the data points used for its training. Motivated by those practices, a large body of recent theoretical research has been devoted to studying…

We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density. These confidence sets are adaptive in $L^2$ loss over a continuous class of Sobolev type spaces. Adaptation holds in…

Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and…

Solving inverse problems governed by partial differential equations (PDEs) is central to science and engineering, yet remains challenging when measurements are sparse, noisy, or when the underlying coefficients are high-dimensional or…

In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev…

This paper is concerned with exponentially ill-posed operator equations with additive impulsive noise on the right hand side, i.e. the noise is large on a small part of the domain and small or zero outside. It is well known that Tikhonov…

A widely believed explanation for the remarkable generalization capacities of overparameterized neural networks is that the optimization algorithms used for training induce an implicit bias towards benign solutions. To grasp this…

A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…

We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_{\Omega}f\left(x, Dv(x)\right)dx\,:\, v\in \mathcal{K}_\psi(\Omega)\right\}, \end{equation*}…

This study investigates the variational posterior convergence rates of inverse problems for partial differential equations (PDEs) with parameters in Besov spaces $B_{pp}^\alpha$ ($p \geq 1$) which are modeled naturally in a Bayesian manner…

Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such…

Solving inverse problems, such as parameter estimation and optimal control, is a vital part of science. Many experiments repeatedly collect data and rely on machine learning algorithms to quickly infer solutions to the associated inverse…

In this paper, we propose to use the general $L^2$-based Sobolev norms, i.e., $H^s$ norms where $s\in \mathbb{R}$, to measure the data discrepancy due to noise in image processing tasks that are formulated as optimization problems. As…

Ridgeless regression has garnered attention among researchers, particularly in light of the ``Benign Overfitting'' phenomenon, where models interpolating noisy samples demonstrate robust generalization. However, kernel ridgeless regression…

Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis…

Uncertainty quantification in PDE inverse problems is essential in many applications. Scientific machine learning and AI enable data-driven learning of model components while preserving physical structure, and provide the scalability and…

We introduce a regularization loss based on kernel mean embeddings with rotation-invariant kernels on the hypersphere (also known as dot-product kernels) for self-supervised learning of image representations. Besides being fully competitive…