奇异偏微分方程的深度学习：一种加权神经网络方法
Deep Learning for Singular PDEs: A Weighted Neural Network Approach
AI总结：
提出一种结合硬边界约束、Softplus正性保持和奇异感知加权残差的深度学习框架，用于数值求解弱奇异半线性椭圆方程，并通过数值实验验证其精度与收敛性。
AI中文摘要：
我们提出了一种深度学习框架，用于数值逼近具有齐次Dirichlet边界条件和正性约束的弱奇异半线性椭圆方程。这些问题具有挑战性，因为当正解在边界附近趋近于零时，非线性源项变得无界。所提出的方法结合了三个组成部分。首先，通过硬约束公式将Dirichlet边界条件直接嵌入到神经表示中。其次，Softplus变换保持计算域内部的积极性，确保训练期间奇异非线性项保持良好定义。第三，奇异感知的加权残差损失强调预测解变小且奇异行为更明显的区域。该方法在弱奇异机制下通过数值参考解和受控验证测试进行评估。与标准残差公式的比较用于评估逼近精度和收敛行为。数值实验展示了将硬边界强制、积极性保持和奇异感知残差加权相结合用于弱奇异椭圆问题的潜力。
英文摘要：
We propose a deep learning framework for the numerical approximation of weakly singular semilinear elliptic equations subject to homogeneous Dirichlet boundary conditions and positivity constraints. These problems are challenging because the nonlinear source term becomes unbounded as the positive solution approaches zero near the boundary. The proposed approach combines three components. First, the Dirichlet boundary conditions are embedded directly into the neural representation through a hard-constraint formulation. Second, a Softplus transformation preserves positivity in the interior of the computational domain, ensuring that the singular nonlinear term remains well-defined during training. Third, a singularity-aware weighted residual loss emphasizes regions where the predicted solution becomes small and the singular behavior is more pronounced. The method is evaluated in the weak singularity regime through numerical reference solutions and controlled verification tests. Comparisons with the standard residual formulation are used to assess approximation accuracy and convergence behavior. The numerical experiments illustrate the potential of combining hard boundary enforcement, positivity preservation, and singularity-aware residual weighting for weakly singular elliptic problems.
