用迭代方法求解非变分型椭圆方程，精度优于现有神经网络方法。
An Iterative Deep Ritz Method for Monotone Elliptic Problems
- 基于迭代优化与凸惩罚，结合函数空间几何信息提升求解器性能
- 在无强正则性假设下实现高精度求解，收敛速率可证明
- 适合处理复杂几何和非标准椭圆问题的科研人员使用
本文提出一种新的迭代深度瑞兹方法（IDRM），用于求解一类广义椭圆问题。该方法受神经网络训练中迭代优化启发，每步结合函数空间几何结构并引入凸惩罚项以增强性能。适用于含单调算子的椭圆问题（不要求变分形式），且对解无严格正则性要求。相比物理信息神经网络和深度瑞兹方法，该方法在同类问题上显著提升精度。通过巴拿赫空间几何与单调算子理论，建立了收敛速率，并分析了学习误差。通过多个挑战性实例验证了方法有效性，包括与现有技术的对比实验。
原文摘要 · Abstract (English)
In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.
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