一种用于模拟奇异摄动问题解的简单浅层神经网络

中文摘要 AI 辅助

我们考虑（前馈）神经网络（NNs）用于模拟带有两个小参数的奇异摄动二阶边值问题的解。我们描述了一种浅层NN，它利用了可用的解的渐近展开式。这些分解为光滑分量和层分量的加性分解，允许得到关于微分阶数以及奇异摄动参数的显式导数估计。利用这类分解，我们提出了一种简单的NN，使用双曲正切（$\tanh$）激活函数结合不同的训练目标（如残差最小化或能量最小化）来模拟这类问题的解。核心思路是在近似空间中加入合适的指数函数，类似有限元方法中的富集空间。一维和二维的数值算例（包括光滑非张量积域）表明，在测试的摄动范围内，该方法具有参数鲁棒性。

英文摘要

We consider (feed-forward) Neural Networks (NNs) for the emulation of the solution to singularly perturbed second order boundary value problems, with two small parameters. We describe a shallow NN which exploits available asymptotic expansions for the solution. These additive decompositions into smooth and layer components, allow for derivative estimates which are explicit in the order of differentiation as well as the singular perturbation parameter(s) \cite{melenk, Irene, SX}. Utilizing such decompositions, we propose a simple NN for emulating the solution to such problems using the $\tanh$ activation function together with different training objectives, such as residual or energy minimization. The key idea is to augment the approximation space with suitable exponential functions, similar to enriched spaces in finite element methods, e.g.~\cite{Kellogg}. Numerical examples in one and two dimensions, including a smooth non-tensor-product domain, illustrate the resulting parameter-robust behavior over the tested perturbation ranges.

发表机构

• Department of Mathematics and Statistics, University of Cyprus（塞浦路斯大学数学与统计系）

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