Machine learning moment closure models for the radiative transfer equation. I: Directly learning a gradient based closure. (English) Zbl 1522.65143
Summary: In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for the radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. A variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 85A25 | Radiative transfer in astronomy and astrophysics |
| 80A21 | Radiative heat transfer |
| 68Q32 | Computational learning theory |
| 68T07 | Artificial neural networks and deep learning |
| 68T09 | Computational aspects of data analysis and big data |
| 82C70 | Transport processes in time-dependent statistical mechanics |
| 35R09 | Integro-partial differential equations |
| 35Q20 | Boltzmann equations |
Keywords:
radiative transfer equation; moment closure; slab geometry; machine learning; neural networkSoftware:
PyTorchReferences:
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