Abstract
We study the performance of long short-term memory networks (LSTMs) and neural ordinary differential equations (NODEs) in learning latent-space representations of dynamical equations for an advection-dominated problem given by the viscous Burgers equation. Our formulation is devised in a nonintrusive manner with an equation-free evolution of dynamics in a reduced space with the latter being obtained through a proper orthogonal decomposition. In addition, we leverage the sequential nature of learning for both LSTMs and NODEs to demonstrate their capability for closure in systems that are not completely resolved in the reduced space. We assess our hypothesis for two advection-dominated problems given by the viscous Burgers equation. We observe that both LSTMs and NODEs are able to reproduce the effects of the absent scales for our test cases more effectively than does intrusive dynamics evolution through a Galerkin projection. This result empirically suggests that time-series learning techniques implicitly leverage a memory kernel for coarse-grained system closure as is suggested through the Mori–Zwanzig formalism.
| Original language | English |
|---|---|
| Article number | 132368 |
| Journal | Physica D: Nonlinear Phenomena |
| Volume | 405 |
| DOIs | |
| State | Published - Apr 2020 |
| Externally published | Yes |
Funding
This material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, USA, Office of Advanced Scientific Computing Research, USA, under Contract DE-AC02-06CH11357. This research was funded in part and used resources of the Argonne Leadership Computing Facility, which is a DOE, USA Office of Science User Facility supported under Contract DE-AC02-06CH11357. This project was also funded by the Los Alamos National Laboratory, USA, 2019 LDRD grant “Machine Learning for Turbulence”. Los Alamos National Laboratory, USA is operated by Triad National Security, LLC, for the National Nuclear Security Administration of US Department of Energy (Contract No. 89233218CNA000001). This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. DOE or the United States Government. This material is based upon work supported by the U.S. Department of Energy (DOE) , Office of Science, USA , Office of Advanced Scientific Computing Research, USA , under Contract DE-AC02-06CH11357 . This research was funded in part and used resources of the Argonne Leadership Computing Facility, which is a DOE, USA Office of Science User Facility supported under Contract DE-AC02-06CH11357 . This project was also funded by the Los Alamos National Laboratory, USA , 2019 LDRD grant “Machine Learning for Turbulence”. Los Alamos National Laboratory, USA is operated by Triad National Security, LLC, for the National Nuclear Security Administration of US Department of Energy (Contract No. 89233218CNA000001 ). This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. DOE or the United States Government.
| Funders | Funder number |
|---|---|
| U.S. Department of Energy | |
| Office of Science | |
| National Nuclear Security Administration | 89233218CNA000001 |
| Advanced Scientific Computing Research | DE-AC02-06CH11357 |
| Laboratory Directed Research and Development | |
| Los Alamos National Laboratory |
Keywords
- Closures
- LSTMs
- Neural ODEs
- ROMs