FOSLS-deRhaNN：用于H(div)和H(curl)的原生de Rham神经类，及其在一阶系统最小二乘神经网络方法求解偏微分方程中的应用
FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equations
- City University of Hong Kong（香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了H(div)和H(curl)的原生de Rham神经类，无需网格，并据此提出FOSLS-deRhaNN方法，以最小二乘泛函在自然空间中求解椭圆、curl-curl及输运方程，误差与泛函等价。
AI中文摘要：
我们构造了图空间H(div)和H(curl)的原生神经逼近类，适用于二维和三维情形，且对于H(div)适用于任意维度。每个实现对于所有参数值都位于该空间中，并且对于具有折点势函数（如ReLU网络）的情况，允许的跳跃以有限宽度出现。这些类是标量网络和分量网络在de Rham复形的固定算子作用下的像，不涉及网格或有限元模拟。对于R^n中的H(div)，我们给出了两个地位相同的原生类，使用反对称势A：Div A + R_n q + h，其中散度q作为显式未知量，以及Div A + z，其中z是H^1场；对于H(curl)，类似的类在二维情形为grad φ + S r + h，在二维和三维情形为grad φ + z。在所有这些类中，场的每个界面跳跃都由势项（Div A或grad φ）承载，而其余部分没有界面跳跃（在正则分解类中它是H^1场）；包含z的类是分量方法并增加了该势项。已知或学习的界面几何通过具有可训练幅度的因子进入势函数，如果散度跳跃，则其余部分也如此。这些类引出了FOSLS-deRhaNN方法，即使用de Rham神经网络的一阶系统最小二乘法，其损失是定义在弱形式的自然空间中的最小二乘泛函；对于椭圆方程，这包括H^{-1}右端项和H^{1/2}Dirichlet数据。具有不连续系数的椭圆方程和curl-curl问题作为实例处理，其泛函等价于误差；具有不连续解的线性输运和具有激波的守恒律使用相同的通量类。
英文摘要：
We construct neural approximation classes native to the graph spaces H(div) and H(curl), in two and three dimensions and, for H(div), in any dimension. Every realization lies in the space for all parameter values, and with kinked potentials, such as ReLU networks, the admissible jumps appear at finite width. The classes are images of scalar and componentwise networks under fixed operators of the de Rham complex, and do not involve a mesh or finite element emulation. For H(div) in R^n two native classes are given on an equal footing, with a skew-symmetric potential $A$: $\mathrm{Div}\,A+R_nq+\mathbf{h}$, with the divergence $q$ as an explicit unknown, and $\mathrm{Div}\,A+\mathbf{z}$ with an $H^1$ field $\mathbf{z}$; for H(curl) the analogous classes are $\mathrm{grad}\,ϕ+Sr+\mathbf{h}$ in two dimensions and $\mathrm{grad}\,ϕ+\mathbf{z}$ in two and three dimensions. In all of them every interface jump of the field is carried by the potential term, $\mathrm{Div}\,A$ or $\mathrm{grad}\,ϕ$, while the remaining part has no interface jump (it is an $H^1$ field in the regular-decomposition classes); the classes with $\mathbf{z}$ are the componentwise approach enriched by this term. Known or learned interface geometry enters the potential through factors with trainable amplitudes, and the remaining part if the divergence jumps. The classes lead to the FOSLS-deRhaNN method, first-order system least squares with de Rham neural networks, whose loss is the least-squares functional posed in the natural spaces of the weak formulation; for elliptic equations this includes $H^{-1}$ right-hand sides and $H^{1/2}$ Dirichlet data. Elliptic equations with discontinuous coefficients and curl-curl problems are treated as instances, with the functional equivalent to the error; linear transport with discontinuous solutions and conservation laws with shocks use the same flux classes.
