Approximation rates of DeepONets for learning operators arising from advection-diffusion equations

被引：45

作者：

Deng, Beichuan ^{[1
]}

Shin, Yeonjong ^{[2
]}

Lu, Lu ^{[3
]}

Zhang, Zhongqiang ^{[1
]}

Karniadakis, George Em ^{[2
]}

机构：

[1] Worcester Polytech Inst, Dept Math Sci, Worcester, MA 01609 USA

[2] Brown Univ, Div Appl Math, Providence, RI USA

[3] Univ Penn, Dept Chem & Biomol Engn, Philadelphia, PA USA

来源：

NEURAL NETWORKS | 2022年 / 153卷

关键词：

Operatorlearning; Fixed-weightneuralnetwork; Advection-diffusion-reactionequations; Burgersequations; ApproximationinBanachspaces; NEURAL-NETWORKS; UNIVERSAL APPROXIMATION; NONLINEAR OPERATORS; UNIFORM APPROXIMATION; FUNCTIONALS;

D O I：

10.1016/j.neunet.2022.06.019

中图分类号：

TP18 [人工智能理论];

学科分类号：

081104 ; 0812 ; 0835 ; 1405 ;

摘要：

We present the analysis of approximation rates of operator learning in Chen and Chen (1995) and Lu et al. (2021), where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection-diffusion equations with or without reaction. We find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.(c) 2022 Elsevier Ltd. All rights reserved.

引用

页码：411 / 426

页数：16

共 42 条

[1]

Ames WF, 1972, NONLINEAR PARTIAL DI

[2] Universal approximation of multiple nonlinear operators by neural networks [J].

Back, AD ;

Chen, TP .

NEURAL COMPUTATION, 2002, 14 (11) :2561-2566

[3] A unified deep artificial neural network approach to partial differential equations in complex geometries [J].

Berg, Jens ;

Nystrom, Kaj .

NEUROCOMPUTING, 2018, 317 :28-41

[4] APPROXIMATIONS OF CONTINUOUS FUNCTIONALS BY NEURAL NETWORKS WITH APPLICATION TO DYNAMIC-SYSTEMS [J].

CHEN, TP ;

CHEN, H .

IEEE TRANSACTIONS ON NEURAL NETWORKS, 1993, 4 (06) :910-918

[5] A unified approach for neural network-like approximation of non-linear functionals [J].

Chen, TP .

NEURAL NETWORKS, 1998, 11 (06) :981-983

[6] UNIVERSAL APPROXIMATION TO NONLINEAR OPERATORS BY NEURAL NETWORKS WITH ARBITRARY ACTIVATION FUNCTIONS AND ITS APPLICATION TO DYNAMICAL-SYSTEMS [J].

CHEN, TP ;

CHEN, H .

IEEE TRANSACTIONS ON NEURAL NETWORKS, 1995, 6 (04) :911-917

[7] APPROXIMATION CAPABILITY TO FUNCTIONS OF SEVERAL VARIABLES, NONLINEAR FUNCTIONALS, AND OPERATORS BY RADIAL BASIS FUNCTION NEURAL NETWORKS [J].

CHEN, TP ;

CHEN, H .

IEEE TRANSACTIONS ON NEURAL NETWORKS, 1995, 6 (04) :904-910

[8] GENERATING EXACT-SOLUTIONS OF THE TWO-DIMENSIONAL BURGERS EQUATIONS [J].

FLETCHER, CAJ .

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, 1983, 3 (03) :213-216

[9]

Guss W. H., 2019, UNIVERSAL APPROXIMAT

[10] RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS [J].

He, Juncai ;

Li, Lin ;

Xu, Jinchao ;

Zheng, Chunyue .

JOURNAL OF COMPUTATIONAL MATHEMATICS, 2020, 38 (03) :502-527

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