椭圆界面问题的一种混合迭代深度Ritz方法
A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems
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中文总结 AI 辅助
提出混合迭代深度Ritz方法，基于混合形式和水平集神经网络求解椭圆界面问题，避免界面采样，数值实验表明在高维、复杂界面和低正则性问题上优于现有神经求解器。
中文摘要 AI 辅助
本文针对二阶椭圆算子的一类界面问题，提出了一种混合迭代深度Ritz方法（H-IDRM）。该方法基于问题的一种新混合形式，并涉及求解一系列凸最小化问题。我们采用水平集神经网络架构，通过界面的水平集表示，以适应解和通量的分段光滑性。该方法仅涉及体积表示，而非界面上的对偶配对，并避免了对复杂界面几何进行显式界面采样的不便。此外，我们对该方法进行了分析，包括神经网络逼近、蒙特卡洛逼近、迭代格式和惩罚参数引起的误差。数值实验表明，H-IDRM在高维域、复杂界面几何和较低子域正则性的问题上优于现有的神经求解器。
英文摘要
In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.
发表机构
- The Chinese University of Hong Kong（香港中文大学）
- Shanghai Normal University（上海师范大学）
机构由 AI 辅助整理，请以论文原文为准。
