Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916138771808256 |
|---|---|
| author | Lee, Jae Yong Schotthöfer, Steffen Xiao, Tianbai Krumscheid, Sebastian Frank, Martin |
| author_facet | Lee, Jae Yong Schotthöfer, Steffen Xiao, Tianbai Krumscheid, Sebastian Frank, Martin |
| contents | This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate the capabilities of the Deep Operator network (DeepONet) approach to modelling the high dimensional collision operator of the linear kinetic equation. This integral operator has crucial analytical structures that a surrogate model, e.g., a DeepONet, needs to preserve to enable meaningful physical simulation. We propose several DeepONet modifications to encapsulate essential structural properties of this integral operator in a DeepONet model. To be precise, we adapt the architecture of the trunk-net so the DeepONet has the same collision invariants as the theoretical kinetic collision operator, thus preserving conserved quantities, e.g., mass, of the modeled many-particle system. Further, we propose an entropy-inspired data-sampling method tailored to train the modified DeepONet surrogates without requiring an excessive expensive simulation-based data generation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16613 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations Lee, Jae Yong Schotthöfer, Steffen Xiao, Tianbai Krumscheid, Sebastian Frank, Martin Numerical Analysis This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate the capabilities of the Deep Operator network (DeepONet) approach to modelling the high dimensional collision operator of the linear kinetic equation. This integral operator has crucial analytical structures that a surrogate model, e.g., a DeepONet, needs to preserve to enable meaningful physical simulation. We propose several DeepONet modifications to encapsulate essential structural properties of this integral operator in a DeepONet model. To be precise, we adapt the architecture of the trunk-net so the DeepONet has the same collision invariants as the theoretical kinetic collision operator, thus preserving conserved quantities, e.g., mass, of the modeled many-particle system. Further, we propose an entropy-inspired data-sampling method tailored to train the modified DeepONet surrogates without requiring an excessive expensive simulation-based data generation. |
| title | Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2402.16613 |