相关论文: Universal Approximation of Nonlinear Operators and…
In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],…
The irreducible unitary representations of the Banach Lie group $U_0(\H)$ (which is the norm-closure of the inductive limit $\cup_k U(k)$) of unitary operators on a separable Hilbert space $\H$, which were found by Kirillov and Ol'shanskii,…
We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which…
Deep neural networks (DNNs) have recently emerged as effective tools for approximating solution operators of partial differential equations (PDEs) including evolutionary problems. Classical numerical solvers for such PDEs often face…
Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain…
The classical universal approximation (UA) theorem for neural networks establishes mild conditions under which a feedforward neural network can approximate a continuous function $f$ with arbitrary accuracy. A recent result shows that neural…
Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved…
Abundant real-world data can be naturally represented by large-scale networks, which demands efficient and effective learning algorithms. At the same time, labels may only be available for some networks, which demands these algorithms to be…
Optimal control problems with nonsmooth objectives and nonlinear partial differential equation (PDE) constraints are challenging, mainly because of the underlying nonsmooth and nonconvex structures and the demanding computational cost for…
Dilation theory is a paradigm for studying operators by way of exhibiting an operator as a compression of another operator which is in some sense well behaved. For example, every contraction can be dilated to (i.e., is a compression of) a…
A plentitude of applications in scientific computing requires the approximation of mappings between Banach spaces. Recently introduced Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) can provide this functionality. For…
We define the concept of higher order differential operators on a general noncommutative, nonassociative superalgebra A, and show that a vertex operator superalgebra has plenty of them, namely modes of vertex operators. A linear operator…
Deep Operator Networks (DeepOnets) have revolutionized the domain of scientific machine learning for the solution of the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space…
We propose a systematic construction of native Banach spaces for general spline-admissible operators ${\rm L}$. In short, the native space for ${\rm L}$ and the (dual) norm $\|\cdot\|_{\mathcal{X}'}$ is the largest space of functions $f:…
We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal…
In this paper, we study {\it operator spaces\/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed subspace $E\subset B(H)$, with $H$ Hilbert. We will be…
Computer vision (CV) is one of the most crucial fields in artificial intelligence. In recent years, a variety of deep learning models based on convolutional neural networks (CNNs) and Transformers have been designed to tackle diverse…
Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff…
Neural operators are capable of capturing nonlinear mappings between infinite-dimensional functional spaces, offering a data-driven approach to modeling complex functional relationships in classical density functional theory (cDFT). In this…
Dynamical Lie algebras (DLAs) have emerged as a valuable tool in the study of parameterized quantum circuits, helping to characterize both their expressiveness and trainability. In particular, the absence or presence of barren plateaus…