Related papers: Neural and spectral operator surrogates: unified c…
We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain $D_{ref}$. The pullback…
We study the approximation rates of a class of deep neural network approximations of operators which arise as data-to-solution maps $\mathcal{S}$ of linear elliptic partial differential equations (PDEs), and act between pairs $X,Y$ of…
We establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or H\"older) continuous maps $\mathcal G:\mathcal X\to\mathcal Y$ between (subsets of) separable Hilbert spaces…
We study the derivative-informed learning of nonlinear operators between infinite-dimensional separable Hilbert spaces by neural networks. Such operators can arise from the solution of partial differential equations (PDEs), and are used in…
This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales.…
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
Learning operators between infinitely dimensional spaces is an important learning task arising in wide applications in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric…
We consider the problem of constructing surrogate operators for parameter-to-solution maps arising from parametric partial differential equations, where repeated forward model evaluations are computationally expensive. We present a…
Neural networks have promise as surrogate partial differential equation (PDE) solvers, but it remains a challenge to use these concepts to solve problems with high accuracy and scalability. In this work, we show that neural network…
The term `surrogate modeling' in computational science and engineering refers to the development of computationally efficient approximations for expensive simulations, such as those arising from numerical solution of partial differential…
We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical…
multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Nonzero multiplication operators on $L^2$ spaces of functions are never compact and then such…
Building surrogate models with uncertainty quantification capabilities is essential for many engineering applications where randomness, such as variability in material properties, is unavoidable. Polynomial Chaos Expansion (PCE) is widely…
Deep Operator Networks are emerging as fundamental tools among various neural network types to learn mappings between function spaces, and have recently gained attention due to their ability to approximate nonlinear operators. In…
In this paper we derive novel families of inclusion sets for the spectrum and pseudospectrum of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or…
We present a framework for automatically structuring and training fast, approximate, deep neural surrogates of stochastic simulators. Unlike traditional approaches to surrogate modeling, our surrogates retain the interpretable structure and…
Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical…
Given a bounded linear operator $T$ on separable Hilbert space, we develop an approach allowing one to construct a matrix representation for $T$ having certain specified algebraic or asymptotic structure. We obtain matrix representations…
Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization, model-based control, and large-scale inverse problems. Surrogate modeling techniques seek to…
State-of-the-art computer codes for simulating real physical systems are often characterized by a vast number of input parameters. Performing uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost always infeasible…