支持面向硬件的深度偏微分方程求解器的神经极弱形式
Neural Very Weak Formulations enabling Hardware-Oriented deep PDE solvers
浏览论文内容
中文总结 AI 辅助
研究椭圆问题最小二乘极弱形式的离散化，核心方法是用低正则性神经网络，测试函数来自适当光滑空间，避免自动微分，在多种情况表现良好，关注特定试验函数利于硬件实现。
中文摘要 AI 辅助
作为概念验证，我们表明，只要测试函数来自适当光滑的空间，椭圆问题的最小二乘极弱形式可以由低正则性的神经网络有效地离散化。除了避免自动微分带来的直接计算优势外，这种方法在各种神经网络空间中评估，即使在奇异解和高维设置等具有挑战性的情况下也表现出良好性能。特别关注基于阶跃激活和一位量化线性函数的试验函数，它们适用于高效的面向硬件的实现。
英文摘要
We show, as a proof of concept, that least-squares very weak formulations of elliptic problems can be effectively discretized by neural networks possessing low regularity, provided the test functions are drawn from appropriately smooth spaces. Apart from the immediate computational benefit of avoiding automatic differentiation, this approach, evaluated across various neural network spaces, demonstrates good performance even in challenging contexts, such as singular solutions and high dimensional settings. Particular attention is paid to trial functions based on step activations and one bit quantized linear functions, which are amenable to efficient hardware-oriented implementations.
