Enforcing exact boundary and initial conditions in the deep mixed residual method Article Swipe

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Liyao Lyu , Keke Wu , Rui Du , Jingrun Chen ·

YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2008.01491

In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically in a systematic manner. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.

Related Topics

Partial Differential Equation

Finite Element Method

Mathematics

Dirichlet Distribution

Computer Science

Mathematical Analysis

Concepts

Residual Boundary value problem Partial differential equation Neumann boundary condition Applied mathematics Boundary (topology) Dirichlet boundary condition Function (biology) Finite element method Mathematics Robin boundary condition Dirichlet distribution Computer science Mathematical analysis Algorithm Physics Biology Evolutionary biology Thermodynamics

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2008.01491
PDF: https://arxiv.org/pdf/2008.01491
OA Status: green
Cited By: 10
References: 12
Related Works: 10
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DOI: https://doi.org/10.48550/arxiv.2008.01491

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Title: Enforcing exact boundary and initial conditions in the deep mixed residual method

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Language: en

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Publication year: 2020

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Publication date: 2020-08-04

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Authors: Liyao Lyu, Keke Wu, Rui Du, Jingrun Chen

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Landing page: https://arxiv.org/abs/2008.01491

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Concepts: Residual, Boundary value problem, Partial differential equation, Neumann boundary condition, Applied mathematics, Boundary (topology), Dirichlet boundary condition, Function (biology), Finite element method, Mathematics, Robin boundary condition, Dirichlet distribution, Computer science, Mathematical analysis, Algorithm, Physics, Biology, Evolutionary biology, Thermodynamics

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Cited by: 10

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abstract_inverted_index.hyperbolic	152
abstract_inverted_index.inevitably	75
abstract_inverted_index.literature	62
abstract_inverted_index.situation,	59
abstract_inverted_index.systematic	108
abstract_inverted_index.approximate	168
abstract_inverted_index.demonstrate	95
abstract_inverted_index.facilitates	172
abstract_inverted_index.Numerically,	16
abstract_inverted_index.consequence,	112
abstract_inverted_index.differential	13
abstract_inverted_index.automatically	105
abstract_inverted_index.wellposedness	10
abstract_inverted_index.high-dimensional	53
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	4
citation_normalized_percentile