This paper studies the problem of learning an unknown function $f$ from given data about $f$. The learning problem is to give an approximation $\hat f$ to $f$ that predicts the values of $f$ away from the data. There are numerous settings for this learning problem depending on (i) what additional information we have about $f$ (known as a model class assumption), (ii) how we measure the accuracy of how well $\hat f$ predicts $f$, (iii) what is known about the data and data sites, (iv) whether the data observations are polluted by noise. A mathematical description of the optimal performance possible (the smallest possible error of recovery) is known in the presence of a model class assumption. Under standard model class assumptions, it is shown in this paper that a near optimal $\hat f$ can be found by solving a certain discrete over-parameterized optimization problem with a penalty term. Here, near optimal means that the error is bounded by a fixed constant times the optimal error. This explains the advantage of over-parameterization which is commonly used in modern machine learning. The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation $\hat f$ of the function $f$ from which the data is collected. Quantitative bounds are given for how much over-parameterization needs to be employed and how the penalization needs to be scaled in order to guarantee a near optimal recovery of $f$. An extension of these results to the case where the data is polluted by additive deterministic noise is also given.

We propose the generative adversarial neural operator (GANO), a generative model paradigm for learning probabilities on infinite-dimensional function spaces. The natural sciences and engineering are known to have many types of data that are sampled from infinite-dimensional function spaces, where classical finite-dimensional deep generative adversarial networks (GANs) may not be directly applicable. GANO generalizes the GAN framework and allows for the sampling of functions by learning push-forward operator maps in infinite-dimensional spaces. GANO consists of two main components, a generator neural operator and a discriminator neural functional. The inputs to the generator are samples of functions from a user-specified probability measure, e.g., Gaussian random field (GRF), and the generator outputs are synthetic data functions. The input to the discriminator is either a real or synthetic data function. In this work, we instantiate GANO using the Wasserstein criterion and show how the Wasserstein loss can be computed in infinite-dimensional spaces. We empirically study GANO in controlled cases where both input and output functions are samples from GRFs and compare its performance to the finite-dimensional counterpart GAN. We empirically study the efficacy of GANO on real-world function data of volcanic activities and show its superior performance over GAN.

We consider the point evaluation of the solution to interface problems with geometric uncertainties, where the uncertainty in the obstacle is described by a high-dimensional parameter $\boldsymbol{y}\in[-1,1]^d$, $d\in\mathbb{N}$. We focus in particular on an elliptic interface problem and a Helmholtz transmission problem. Point values of the solution in the physical domain depend in general non-smoothly on the high-dimensional parameter, posing a challenge when one is interested in building surrogates. Indeed, high-order methods show poor convergence rates, while methods which are able to track discontinuities usually suffer from the so-called curse of dimensionality. For this reason, in this work we propose to build surrogates for point evaluation using deep neural networks. We provide a theoretical justification for why we expect neural networks to provide good surrogates. Furthermore, we present extensive numerical experiments showing their good performance in practice. We observe in particular that neural networks do not suffer from the curse of dimensionality, and we study the dependence of the error on the number of point evaluations (that is, the number of discontinuities in the parameter space), as well as on several modeling parameters, such as the contrast between the two materials and, for the Helmholtz transmission problem, the wavenumber.

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with $\ell^1$ norm, the $p$-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. A new class of Orlicz RKBSs is proposed. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs.

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.

The purpose of this paper is to establish the notion of vector-valued reproducing kernel Banach spaces and demonstrate its applications to multi-task machine learning. Built on the theory of scalar-valued reproducing kernel Hilbert spaces (RKHS) [3], kernel methods have been proven successful in single task machine learning [10, 14, 29, 30, 33]. Multi-task learning where the unknown target function to be learned from finite sample data is vector-valued appears more often in practice. References [13, 25] proposed the development of kernel methods for learning multiple related tasks simultaneously. The mathematical foundation used there was the theory of vector-valued RKHS [5, 27].