- Title Information
- Title
- Galerkin neural networks for the approximation of partial differential equations with error control
- Type of Resource (primo)
- dissertations
- Name:
Personal
- Name Part
- Dong, Justin
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Ainsworth, Mark
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Keith, Brendan
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Stinis, Panagiotis
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2023
- Physical Description
- Extent
- 21, 179 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2023
- Genre (aat)
- theses
- Abstract
- In the past decade, deep neural networks have seen a dramatic rise in popularity for a wide variety of tasks from computer vision and speech recognition to drug discovery. They have even enjoyed increasing use in fields that have traditionally been within the purview of computational scientists and mathematicians, chief among them the simulation of differential equations that model our physical world. And yet, despite the many successes of methodologies based on neural networks for approximating differential equations, their behavior is still not well understood and little theory exists to substantiate the convergence of such methods.
In this work, we introduce a new method -- Galerkin neural networks -- for approximating differential equations in variational form which allows for rigorous control of the approximation error. The basic idea of the method is to project the solution onto a finite-dimensional subspace whose basis functions are realizations of coarse neural networks. These basis functions are approximate Riesz representations of weak residuals of the variational equation and comprise a sequence of corrections to a given approximation of the solution. We prove error estimates for the method and demonstrate that the learned basis functions naturally induce an a posteriori error estimator when paired with the weak residual. Applications to solid mechanics and fluid mechanics are presented, including Reissner-Mindlin plates, channel-driven cavity flows, and suspension Poiseuille flows which demonstrate the ability of the method capture singular and low-regularity features and complex multiscale behavior such as boundary layers.
- Subject
- Topic
- Neural Networks
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00893488")
- Topic
- Differential equations, Partial--Numerical solutions
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20230602