标题： 基于自适应基的深度神经网络用于求解具有局部特征的偏微分方程 显示英文标题

标题： Adaptive Basis-inspired Deep Neural Network for Solving Partial Differential Equations with Localized Features

摘要： 本文提出了一种基于自适应基函数的深度神经网络（ABI-DNN），用于求解具有局部现象（如陡峭梯度和奇异性）的偏微分方程。 类似于自适应有限元方法， ABI-DNN 包含一个“求解、估计、标记、增强”的迭代过程，可以自动识别困难区域并添加新的神经元以增强其能力。 一个关键挑战是，在对它们在逼近中的作用理解有限的情况下，迫使新神经元集中在已识别的区域。 为了解决这个问题，我们受到有限元基函数的启发，构建了新颖的基函数启发块（BI-block），以帮助理解每个块的贡献。 在 BI-block 和著名的柯尔莫哥洛夫叠加定理的帮助下，我们首先开发了一种名为基函数启发深度神经网络（BI-DNN）的新固定网络架构，然后将其集成到上述自适应框架中，提出了 ABI-DNN。 大量的数值实验表明，BI-DNN 和 ABI-DNN 都能有效捕捉目标函数中的挑战性奇异性。 与 PINN 相比，BI-DNN 在可训练参数数量相似的情况下，获得了显著更低的相对误差。 当设置指定的容差时，ABI-DNN 可以自适应地学习一个适当的架构，其误差与具有相同结构的 BI-DNN 的误差相当。 显示英文摘要

摘要： This paper proposes an Adaptive Basis-inspired Deep Neural Network (ABI-DNN) for solving partial differential equations with localized phenomena such as sharp gradients and singularities. Like the adaptive finite element method, ABI-DNN incorporates an iteration of "solve, estimate, mark, enhancement", which automatically identifies challenging regions and adds new neurons to enhance its capability. A key challenge is to force new neurons to focus on identified regions with limited understanding of their roles in approximation. To address this, we draw inspiration from the finite element basis function and construct the novel Basis-inspired Block (BI-block), to help understand the contribution of each block. With the help of the BI-block and the famous Kolmogorov Superposition Theorem, we first develop a novel fixed network architecture named the Basis-inspired Deep Neural Network (BI-DNN), and then integrate it into the aforementioned adaptive framework to propose the ABI-DNN. Extensive numerical experiments demonstrate that both BI-DNN and ABI-DNN can effectively capture the challenging singularities in target functions. Compared to PINN, BI-DNN attains significantly lower relative errors with a similar number of trainable parameters. When a specified tolerance is set, ABI-DNN can adaptively learn an appropriate architecture that achieves an error comparable to that of BI-DNN with the same structure.