TY - JOUR

T1 - Finite neuron method and convergence analysis

AU - Xu, Jinchao

N1 - Funding Information:
Main results in this manuscript were prepared for and reported in the “International Conference on Computational Mathematics and Scientific Computing” (August 17-20, 2020, http://lsec.cc.ac.cn/∼iccmsc/Home.html), and the author is grateful to the conference organizers for their invitation to present and also to the audience for their helpful feedback. The author also wishes to thank Limin Ma, Qingguo Hong and Shuo Zhang for their help in preparing this manuscript and Jonathan Siegel and Juncai He for helpful discussions. This work was partially supported by the Verne M. William Professorship Fund from Penn State University and the National Science Foundation (Grant No. DMS-1819157).
Publisher Copyright:
© 2020 Global-Science Press
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/11

Y1 - 2020/11

N2 - We study a family of Hm-conforming piecewise polynomials based on the artificial neural network, referred to as the finite neuron method (FNM), for numerical solution of 2m-th-order partial differential equations in Rd for any m,d ≥ 1 and then provide convergence analysis for this method. Given a general domain Ω ⊂ Rd and a partition Th of Ω, it is still an open problem in general how to construct a conforming finite element subspace of Hm(Ω) that has adequate approximation properties. By using techniques from artificial neural networks, we construct a family of Hm-conforming functions consisting of piecewise polynomials of degree k for any k ≥ m and we further obtain the error estimate when they are applied to solve the elliptic boundary value problem of any order in any dimension. For example, the error estimates that ku−uNkHm(Ω) = O(N− 21 − 1d ) is obtained for the error between the exact solution u and the finite neuron approximation uN. A discussion is also provided on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for the finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can be obtained by only solving a non-linear and non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requires further investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and the convenience of the reader, some basic known results and their proofs.

AB - We study a family of Hm-conforming piecewise polynomials based on the artificial neural network, referred to as the finite neuron method (FNM), for numerical solution of 2m-th-order partial differential equations in Rd for any m,d ≥ 1 and then provide convergence analysis for this method. Given a general domain Ω ⊂ Rd and a partition Th of Ω, it is still an open problem in general how to construct a conforming finite element subspace of Hm(Ω) that has adequate approximation properties. By using techniques from artificial neural networks, we construct a family of Hm-conforming functions consisting of piecewise polynomials of degree k for any k ≥ m and we further obtain the error estimate when they are applied to solve the elliptic boundary value problem of any order in any dimension. For example, the error estimates that ku−uNkHm(Ω) = O(N− 21 − 1d ) is obtained for the error between the exact solution u and the finite neuron approximation uN. A discussion is also provided on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for the finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can be obtained by only solving a non-linear and non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requires further investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and the convenience of the reader, some basic known results and their proofs.

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U2 - 10.4208/CICP.OA-2020-0191

DO - 10.4208/CICP.OA-2020-0191

M3 - Article

AN - SCOPUS:85097453560

VL - 28

SP - 1707

EP - 1745

JO - Communications in Computational Physics

JF - Communications in Computational Physics

SN - 1815-2406

IS - 5

ER -