相关论文: Wiener Chaos Expansion based Neural Operator for S…
We treat some classes of linear and semilinear stochastic partial differential equations of Schr\"odinger type on $\mathbb{R}^d$, involving a non-flat Laplacian, within the framework of white noise analysis, combined with Wiener-It\^o chaos…
Driven by rapid advances in artificial intelligence and modern GPU computing capabilities, deep learning methods based on the optimization paradigm have provided new pathways to solve spatiotemporal physical problems, whose mathematical…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the…
Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural…
This paper introduces an operator-based neural network, the mirror-padded Fourier neural operator (MFNO), designed to learn the dynamics of stochastic systems. MFNO extends the standard Fourier neural operator (FNO) by incorporating mirror…
In this paper, we present a novel Fredholm Integral Equation Neural Operator (FIE-NO) method, an integration of Random Fourier Features and Fredholm Integral Equations (FIE) into the deep learning framework, tailored for solving data-driven…
Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of…
Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While…
Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…
Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based…
The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution…
Deep operator network (DeepONet) has shown significant promise as surrogate models for systems governed by partial differential equations (PDEs), enabling accurate mappings between infinite-dimensional function spaces. However, when applied…
The fundamental lemma by Jan C. Willems and co-workers, which is deeply rooted in behavioral systems theory, has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style…
Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have…
Neural operators improve conventional neural networks by expanding their capabilities of functional mappings between different function spaces to solve partial differential equations (PDEs). One of the most notable methods is the Fourier…
Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training…
Neural operators have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). However, standard spectral methods based on Fourier transforms struggle with problems involving discontinuous…
Neural operator models for solving partial differential equations (PDEs) often rely on global mixing mechanisms-such as spectral convolutions or attention-which tend to oversmooth sharp local dynamics and introduce high computational cost.…
In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first…