谱巴伦空间中双调和方程的混合残差法
A mixed residual method for biharmonic equations in spectral Barron spaces
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中文总结 AI 辅助
该研究针对非齐次夹紧边界条件的双调和方程，提出混合残差法，通过建立方程在谱巴伦空间的适定性推导误差界，克服维数灾难，其误差界含近似与泛化误差两部分，数值实验验证了方法有效性。
中文摘要 AI 辅助
我们提出一种混合残差法（MIM）来数值求解具有非齐次夹紧边界条件的双调和方程。通过在谱巴伦空间中建立双调和方程的适定性，我们推导了MIM的误差界，该误差界将浅神经网络近似与精确解联系起来并克服了维数灾难。此误差界由两部分组成：第一部分对应神经网络的近似误差，第二部分表示随机采样训练数据产生的泛化误差。我们给出了几个数值实验来证明所提方法的有效性。
英文摘要
We propose a mixed residual method (MIM) for numerically solving the biharmonic equation with nonhomogeneous clamped boundary conditions. By establishing the well-posedness of the biharmonic equation in spectral Barron spaces, we derive an error bound for MIM that relates shallow neural network approximations to the exact solution and overcomes the curse of dimensionality. This error bound consists of two components: the first corresponds to the approximation error of the neural network, while the second represents the generalization error arising from randomly sampled training data. Several numerical experiments are presented to demonstrate the effectiveness of the proposed method.
