Machine learning for computational engineering
- This dissertation presents my work on solving inverse problems in computational science and engineering using a combination of partial differential equations (PDEs) and deep neural networks (DNNs). By substituting unknown parts of a physical model with DNNs and expressing known physical laws with PDEs, we preserve physical structures (e.g., conservation of momentum) to the largest extent while leveraging DNNs for data-driven function approximation. To train the DNNs within a physical system, we express both numerical simulations (e.g., finite element method) and DNNs as computational graphs and calculate the gradients using reverse-mode automatic differentiation. We have built a system of reusable and flexible numerical simulation operators in our software, ADCME, and AdFem, to help users implement sophisticated numerical schemes using a high-level language, Julia, without constructing computational graphs explicitly. Our software has benefited a variety of engineering applications, such as seismic inversion, neural-network-based constitutive modeling, inverse modeling of Navier-Stokes equations, etc. We have also extended ADCME to large-scale inverse modeling using MPI and successfully deployed distributed inverse modeling optimization algorithms across thousands of cores. My major contribution includes: - Algorithms: first order physics constrained learning for calculating gradients of implicit operators, and second order physics constrained learning for calculating Hessians. These techniques find novel applications in coupled systems of PDEs and DNNs. For example, we proposed trust region methods for training deep neural networks embedded in a PDE solver, which enjoy remarkably faster convergence and better accuracy than first order optimizers in our cases. - Software: ADCME, a general automatic differentiation framework for inverse modeling, and AdFem, a computational graph based finite element simulator. We also extended their capability to large-scale problems via MPI. - Applications: many fruitful applications, such as DNN-based constitutive modeling for solid mechanics and geomechanics, calibrating Lévy processes from sample paths, and learning unknown distributions from indirect data in a physical system.
|Type of resource
|electronic resource; remote; computer; online resource
|1 online resource.
|Degree committee member
|Degree committee member
|Stanford University, Institute for Computational and Mathematical Engineering
|Statement of responsibility
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis Ph.D. Stanford University 2021.
- © 2021 by Kailai Xu
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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