Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries
Charalambos Makridakis and Aaron Pim and Tristan Pryer and Nikolaos Rekatsinas
我们提出了一个神经网络框架,用于解决具有流入边界条件的固定线性传输方程。 该方法代表使用神经网络的解决方案,并通过拉格朗日乘数施加边界条件,基于受经典Uzawa算法启发的马鞍点公式。 该计划是无网状的,与自动分化兼容,并自然地延伸到散射和异构介质的问题。 我们建立连续体配方的收敛,并分析二次误差,神经近似和离散实现中不精确的优化的影响。 数字实验表明,该方法捕获各向异的传输,强制执行边界条件并准确解决散射动力学。
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 contin...