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Data-Driven Approaches for Differential Equation Governing Systems.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Data-Driven Approaches for Differential Equation Governing Systems.
Author:	Hu, Yihao.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2022
Description:	130 p.
Notes:	Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
Notes:	Advisor: Xu, Zhiliang.
Contained By:	Dissertations Abstracts International84-01B.
Subject:	Applied mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29164276
ISBN:	9798438799313

Data-Driven Approaches for Differential Equation Governing Systems.
Hu, Yihao.

Data-Driven Approaches for Differential Equation Governing Systems. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 130 p.

Source: Dissertations Abstracts International, Volume: 84-01, Section: B.

Thesis (Ph.D.)--University of Notre Dame, 2022.

This item must not be sold to any third party vendors.

Applying deep learning methods to solve high-dimensional and nonlinear differential equations(DE) has raised much attention recently. A goal of using machine learning in systems of differential equations is to train a surrogate model with prior physics information and generate predictions with stability and accuracy. However, training such models for high-dimensional/nonlinear multi-scale ODE or PDE systems with limited or labeled data is a grant challenge; and the proper design of the architecture of the neural network is still poorly understood.This dissertation explores data-driven methods in modeling and predicting differential equation governing systems. To tackle the training issue in learning switch systems with imbalanced scales, we propose a novel PINN-based neural network model that resolves the training issue of regular PINN in learning nonlinear switch systems. We explore and incorporate batch statistics in physics-constrained loss functions. The numerical results are demonstrated via three examples by semi-supervised learning and supervised learning algorithms with a small batch of the signal dataset.For learning the system of PDEs, a sequence to sequence supervised learning model for PDEs named Neural-PDE is proposed in this work. Unlike the conventional machine learning approaches for learning PDEs, such as CNN and MLP, which require a great number of parameters for model precision, the Neural-PDE utilizes an RNN based structure, which shares parameters among all-time steps. Thus the Neural-PDE considerably reduces computational complexity and leads to a fast learning algorithm. We showcase the prediction power of the Neural-PDE by applying it to problems from 1D PDEs to a multi-scale complex fluid system.Motivated by those innovative methodologies for learning systems of differential equations, we develop a machine learning framework for learning the dynamics of time-dependent oceanic variables across multiply detection sensors. The prediction accurately replicates complex signals and provides comparable performance to state-of-the-art benchmarks.

ISBN: 9798438799313Subjects--Topical Terms:

377601
Applied mathematics.
Subjects--Index Terms:

Differential equations

Data-Driven Approaches for Differential Equation Governing Systems.
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500 $a Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
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506 $a This item must not be sold to any third party vendors.
520 $a Applying deep learning methods to solve high-dimensional and nonlinear differential equations(DE) has raised much attention recently. A goal of using machine learning in systems of differential equations is to train a surrogate model with prior physics information and generate predictions with stability and accuracy. However, training such models for high-dimensional/nonlinear multi-scale ODE or PDE systems with limited or labeled data is a grant challenge; and the proper design of the architecture of the neural network is still poorly understood.This dissertation explores data-driven methods in modeling and predicting differential equation governing systems. To tackle the training issue in learning switch systems with imbalanced scales, we propose a novel PINN-based neural network model that resolves the training issue of regular PINN in learning nonlinear switch systems. We explore and incorporate batch statistics in physics-constrained loss functions. The numerical results are demonstrated via three examples by semi-supervised learning and supervised learning algorithms with a small batch of the signal dataset.For learning the system of PDEs, a sequence to sequence supervised learning model for PDEs named Neural-PDE is proposed in this work. Unlike the conventional machine learning approaches for learning PDEs, such as CNN and MLP, which require a great number of parameters for model precision, the Neural-PDE utilizes an RNN based structure, which shares parameters among all-time steps. Thus the Neural-PDE considerably reduces computational complexity and leads to a fast learning algorithm. We showcase the prediction power of the Neural-PDE by applying it to problems from 1D PDEs to a multi-scale complex fluid system.Motivated by those innovative methodologies for learning systems of differential equations, we develop a machine learning framework for learning the dynamics of time-dependent oceanic variables across multiply detection sensors. The prediction accurately replicates complex signals and provides comparable performance to state-of-the-art benchmarks.
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