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Applications of Machine Learning to Finite Volume Methods


Stevens, Benjamin Carter (2022) Applications of Machine Learning to Finite Volume Methods. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/41qn-7n22.


The finite volume method (FVM) has been one of the primary tools of computational fluid dynamics (CFD) for many decades. This method allows for the approximate solution of a partial differential equation (PDE) to be determined by breaking up a problem with no analytical solution into smaller pieces that can be solved together to get a physically realistic simulation. These algorithms can even be used for PDEs with discontinuous solutions, though they must be carefully designed for those situations because they cannot assume any level of smoothness in the solution. An FVM that has been designed for PDEs with discontinuous solutions is referred to as a shock-capturing method. For most of their history, FVM algorithms have been developed using rigorous mathematical arguments to formally maximize the order of convergence of the solution as the grid is refined. However, these arguments depend on the solution to the PDE being smooth, and therefore do not apply to shock-capturing methods. Instead, shock-capturing methods have traditionally been designed using human intuition to create algorithms that then perform well empirically. In this thesis, we instead follow a data-driven approach to train neural networks to use for enhanced FVM methods.

By including a neural network in our FVM, we can use empirical data to optimize the algorithm. We can also utilize ideas from traditional FVM algorithms to create hybrid methods that have tunable parameters and maintain convergence guarantees present in FVMs that have been designed by hand. We explore these hybrid methods in a variety of settings. First, we create a general-purpose shock-capturing method WENO-NN by hybridizing the popular shock-capturing method WENO-JS with a neural network. Additionally, we develop a network architecture, called FiniteNet, that can be used to learn a coarse-graining model associated with a specific PDE and embed it into an FVM scheme. Finally, we also explore the idea of using transfer learning to further improve the WENO-NN for specific problems and name the resulting algorithm WENO-TL. We demonstrate experimentally that this hybrid approach results in methods that can offer similar error levels as traditional FVMs at less computational cost. Although the neural network increases the computational cost of one evaluation of our hybrid FVM, these methods also allow the simulation to be carried out on a coarser grid, leading to a net reduction in both simulation time and memory usage.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Fluid mechanics, machine learning, finite difference, finite volume
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Mechanical Engineering
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Colonius, Tim
Thesis Committee:
  • Blanquart, Guillaume (chair)
  • Yue, Yisong
  • Anandkumar, Anima
  • Colonius, Tim
Defense Date:23 June 2021
Funding AgencyGrant Number
NSF Graduate Research Fellowship1745301
Record Number:CaltechTHESIS:02062022-203334930
Persistent URL:
Related URLs:
URLURL TypeDescription article adapted for Chapter 3 paper adapted for Chapter 4
Stevens, Benjamin Carter0000-0002-3410-5922
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14489
Deposited By: Benjamin Stevens
Deposited On:12 Mar 2022 00:27
Last Modified:25 Oct 2023 20:36

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