Related papers: Benign overfitting in Fixed Dimension via Physics-…
Physics Informed Neural Networks (PINNs) is a promising application of deep learning. The smooth architecture of a fully connected neural network is appropriate for finding the solutions of PDEs; the corresponding loss function can also be…
Motivated by recent work on benign overfitting in overparameterized machine learning, we study the generalization behavior of functions in Sobolev spaces $W^{k, p}(\mathbb{R}^d)$ that perfectly fit a noisy training data set. Under…
We propose SDORE, a Semi-supervised Deep Sobolev Regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep ReQU neural networks to minimize the empirical risk with gradient norm…
Benign overfitting is a phenomenon in machine learning where a model perfectly fits (interpolates) the training data, including noisy examples, yet still generalizes well to unseen data. Understanding this phenomenon has attracted…
Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a…
In this paper, we present a novel methodology for automatic adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), and we demonstrate that this makes it possible to robustly address multi-objective and multi-scale…
We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the…
The inference performance of the pseudolikelihood method is discussed in the framework of the inverse Ising problem when the $\ell_2$-regularized (ridge) linear regression is adopted. This setup is introduced for theoretically investigating…
An ill-posed inverse problem of autoconvolution type is investigated. This inverse problem occurs in nonlinear optics in the context of ultrashort laser pulse characterization. The novelty of the mathematical model consists in a physically…
Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. In this work, we analyze a new data-driven regularized stochastic gradient descent…
The phenomenon of benign overfitting, where a predictor perfectly fits noisy training data while attaining near-optimal expected loss, has received much attention in recent years, but still remains not fully understood beyond well-specified…
By selecting different filter functions, spectral algorithms can generate various regularization methods to solve statistical inverse problems within the learning-from-samples framework. This paper combines distributed spectral algorithms…
We study the generalization of two-layer ReLU neural networks in a univariate nonparametric regression problem with noisy labels. This is a problem where kernels (\emph{e.g.} NTK) are provably sub-optimal and benign overfitting does not…
A particularly interesting instance of supervised learning with kernels is when each training example is associated with two objects, as in pairwise classification (Brunner et al., 2012), and in supervised learning of preference relations…
The phenomenon of benign overfitting is one of the key mysteries uncovered by deep learning methodology: deep neural networks seem to predict well, even with a perfect fit to noisy training data. Motivated by this phenomenon, we consider…
Neural networks have emerged as powerful surrogates for solving partial differential equations (PDEs), offering significant computational speedups over traditional methods. However, these models suffer from a critical limitation: error…
We develop a kernel-based approach for estimating the spatially varying Sobolev regularity~$s$ of an unknown $d$-variate function~$f$ from scattered sampling data, which quantifies the degree of local differentiability supported by the…
Inverse problems in imaging are typically ill-posed and are usually solved by employing regularized optimization techniques. The usage of appropriate constraints can restrict the solution space, thus making it feasible for a reconstruction…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing partial differential equations (PDEs), to ensure its output is consistent with fundamental physics laws.…