相关论文: Structure-informed operator learning for parabolic…

The deep operator networks (DeepONet), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a…

The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties…

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…

Physics-informed neural networks approach the approximation of differential equations by directly incorporating their structure and given conditions in a loss function. This enables conditions like, e.g., invariants to be easily added…

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with…

The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme}…

Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep…

Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large…

The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…

We present and analyse an approach to image reconstruction problems with imperfect forward models based on partially ordered spaces - Banach lattices. In this approach, errors in the data and in the forward models are described using order…

We propose an approach to solving partial differential equations (PDEs) using a set of neural networks which we call Neural Basis Functions (NBF). This NBF framework is a novel variation of the POD DeepONet operator learning approach where…

A new data-driven method for operator learning of stochastic differential equations(SDE) is proposed in this paper. The central goal is to solve forward and inverse stochastic problems more effectively using limited data. Deep operator…

Neural operator learning models have emerged as very effective surrogates in data-driven methods for partial differential equations (PDEs) across different applications from computational science and engineering. Such operator learning…

We study a new nonlinear system which contains a partial differential equation, a quasivariational inequality and a parabolic variational inequality in Banach spaces. We obtain the unique solvability of the coupled system under moderate…

This article introduces a novel approach to learning monotone neural networks through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone…

Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success,…

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…

We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator…

This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized…

In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first…