Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912502767419392 |
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| author | Makridakis, Charalambos Pim, Aaron Pryer, Tristan Rekatsinas, Nikolaos |
| author_facet | Makridakis, Charalambos Pim, Aaron Pryer, Tristan Rekatsinas, Nikolaos |
| contents | We propose a neural network framework for solving stationary linear transport equations with inflow boundary conditions. The method represents the solution using a neural network and imposes the boundary condition via a Lagrange multiplier, based on a saddle-point formulation inspired by the classical Uzawa algorithm. The scheme is mesh-free, compatible with automatic differentiation and extends naturally to problems with scattering and heterogeneous media. We establish convergence of the continuum formulation and analyse the effects of quadrature error, neural approximation and inexact optimisation in the discrete implementation. Numerical experiments show that the method captures anisotropic transport, enforces boundary conditions and resolves scattering dynamics accurately. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19907 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries Makridakis, Charalambos Pim, Aaron Pryer, Tristan Rekatsinas, Nikolaos Numerical Analysis Optimization and Control Computational Physics We propose a neural network framework for solving stationary linear transport equations with inflow boundary conditions. The method represents the solution using a neural network and imposes the boundary condition via a Lagrange multiplier, based on a saddle-point formulation inspired by the classical Uzawa algorithm. The scheme is mesh-free, compatible with automatic differentiation and extends naturally to problems with scattering and heterogeneous media. We establish convergence of the continuum formulation and analyse the effects of quadrature error, neural approximation and inexact optimisation in the discrete implementation. Numerical experiments show that the method captures anisotropic transport, enforces boundary conditions and resolves scattering dynamics accurately. |
| title | Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries |
| topic | Numerical Analysis Optimization and Control Computational Physics |
| url | https://arxiv.org/abs/2507.19907 |