Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


Dissertation_Zhen_Chen_revised.pdf (3.01 MB)

ETD Abstract Container

Abstract Header

Deep Learning of Unknown Governing Equations

Chen, Zhen
http://rave.ohiolink.edu/etdc/view?acc_num=osu162687741780147

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
For many problems in science and engineering, there are lots of observational, experimental, or simulation data. Governing equations for modeling the underlying dynamics and physical laws hidden in the data are often not fully known for many systems in modern applications. The thesis is concerned with designing machine learning methods to recover/discover unknown governing equations and mathematical models from data. The first part of this thesis focuses on data-driven recovery of unknown dynamical systems. We propose a deep learning method that uses data of the state variables to recover unknown governing equations with embedded unknown/uncertain param- eters. We introduce additional inputs in the deep neural networks (DNN) structure to incorporate the unknown system parameters. This allows us to register system responses with respect to different system parameters. We further develop a method for recovering non-autonomous systems, for which the solution states depend on time- dependent input and the entire history of the system states. The second part of this thesis focuses on model correction using data. We pro- pose a new framework called generalized residual network (gResNet). This framework broadly defines “residue” as the discrepancy between measurement data and predic- tion model by another model, which can be an existing coarse model or reduced order model. In this sense, the gResNet serves as a model correction to the existing model and recovers the unresolved dynamics. We demonstrate that the gResNet is capa- ble of learning the underlying unknown equations and producing predictions with accuracy higher than the standard ResNet structure. The third part of this thesis is devoted to deep learning of partial differential equations (PDEs). We establish a new deep learning framework in nodal space. The data are measurement of the solution states on a set of grids/nodes. Our work conducts the learning directly in physical space by approximating evolution operator of the underlying PDE. To achieve this, we propose a new DNN structure, consisting of a disassembly block and an assembly layer, that has a direct correspondence to a general time-stepping evolution of the unknown PDE. Our DNN model does not rely on any geometric structure of nodal grids. On the practical side, the proposed DNN structure allows one to use structure-free grids/nodes without any geometric information.
Dongbin Xiu (Advisor)
Yulong Xing (Committee Member)
King-Yeung Lam (Committee Member)
Mathematics
Machine Learning, Data-Driven Modelling

Recommended Citations

Citations

  • Chen, Z. (2021). Deep Learning of Unknown Governing Equations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu162687741780147

    APA Style (7th edition)

  • Chen, Zhen. Deep Learning of Unknown Governing Equations. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu162687741780147.

    MLA Style (8th edition)

  • Chen, Zhen. "Deep Learning of Unknown Governing Equations." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu162687741780147

    Chicago Manual of Style (17th edition)

osu162687741780147
145
© 2021, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.