发表机构
Kyung Hee University; Hanyang University; Texas State University（庆熙大学; 汉阳大学; 德州州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种局部守恒的富集线性化神经网络方法求解达西流模型，通过添加分片常数函数和GLS项弥补逆不等式缺失，证明最优H1阶误差估计，数值实验验证了其局部守恒性和高精度。
AI 中文摘要
本文针对达西流模型提出了一种局部守恒的富集线性化神经网络（ENN）方法。一个线性化的浅层ReLU$^k$网络（其隐藏层参数固定于球面的拟均匀集）承担逼近任务，并在区域的辅助细分上添加分片常数函数以确保局部质量守恒。该细分可由一般的、可能弯曲的单元组成。由于神经网络函数不满足逆不等式，标准间断伽辽金内罚格式的稳定性与误差分析不再适用。缺失的逆不等式通过边恒等式结合伽辽金最小二乘（GLS）项来弥补，同时使用一个在$L^2$、$H^1$和$H^2$范数下同时最优的单一神经逼近器。我们采用非对称内罚（NIPG）形式，该形式对任何正罚参数都是强制的。以ENN中使用的浅层神经网络的宽度$n$，我们证明了在EG范数下的误差估计，该估计具有最优的$H^1$阶，即对于$H^r$（$r\ge2$）中的解为$O(n^{-(r-1)/d})$，对于$H^s$（$\frac32<s\le2$）中的解为$O(n^{-(s-1)/d})$，适用于任何形状规则的细分，当GLS参数满足$\min\{h,n^{-1/d}\}^2\lesssim\tau\lesssim n^{-2/d}$且边项按$\min\{h_e,n^{-1/d}\}$缩放时，即按网络的分辨率而非网格的分辨率缩放。构造了一个数值达西通量，该通量无论单元大小或数量如何都是局部守恒的。样本数值结果证明了理论的正确性。特别地，固定的$4\times4$细分与随网络细化的细分具有相同的精度，且质量损失保持在舍入误差水平，而非守恒的神经网络方法则不然。
英文摘要
This paper presents a locally conservative Enriched Linearized Neural Network (ENN) method for the Darcy flow model. A linearized shallow ReLU$^k$ network, whose hidden-layer parameters are fixed on a quasi-uniform set of the sphere, carries the approximation, and piecewise constant functions on an auxiliary subdivision of the domain are added to ensure local mass conservation. The subdivision may consist of general, possibly curved, elements. Since neural network functions do not satisfy the inverse inequality, the stability and error analysis of the standard discontinuous Galerkin interior penalty formulation do not apply. The missing inverse inequality is remedied by an edge identity combined with a Galerkin least-squares (GLS) term, together with a single neural approximant that is optimal in the $L^2$, $H^1$ and $H^2$ norms simultaneously. We use the nonsymmetric interior penalty (NIPG) form, which is coercive for every positive penalty parameter. With $n$, the width of the shallow neural network, used in ENN, we prove an error estimate in the EG norm that is of the optimal $H^1$ order, i.e., $O(n^{-(r-1)/d})$ for solutions in $H^r$, $r\ge2$, and $O(n^{-(s-1)/d})$ for solutions in $H^s$, $\frac32<s\le2$, for any shape-regular subdivision, when the GLS parameter satisfies $\min\{h,n^{-1/d}\}^2\lesssimτ\lesssim n^{-2/d}$ and the edge terms are scaled with $\min\{h_e,n^{-1/d}\}$, i.e., by the resolution of the network rather than of the mesh. A numerical Darcy flux is constructed, which is locally conservative, regardless of the size or the number of the elements. Sample numerical results demonstrate the correctness of the theory. In particular, a fixed $4\times4$ subdivision gives the same accuracy as a subdivision refined with the network, and the mass loss stays at round-off unlike a non-conservative neural network method.
