Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries

TL;DR

This paper introduces a neural network-based method for solving stationary linear transport equations that enforces boundary conditions via a Lagrange multiplier, demonstrating convergence and accuracy in complex transport problems.

Contribution

It presents a novel mesh-free neural network framework using a saddle-point formulation inspired by Uzawa's algorithm for kinetic transport equations.

Findings

The method accurately captures anisotropic transport.

It enforces boundary conditions effectively.

It resolves scattering dynamics with high accuracy.

Abstract

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.