发表机构
Department of Mathematics; Kabale University（数学系; 卡巴勒大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非线性多尺度椭圆方程物理信息学习的统计公式差距，为边界兼容变分神经求解器推导误差界，构造一维周期扩散方程最小障碍见证，通过数值评估得出相关指数，表明变分公式可消除统计惩罚但未解决多尺度逼近问题。
AI 中文摘要
我们证明了非线性多尺度椭圆方程物理信息学习中的有限样本公式差距。对于系数在尺度\(\epsilon\)上振荡的一致单调散度形式类，我们为边界兼容变分神经求解器推导了有限宽度、有限样本和有限迭代误差界。其稳定性、采样和优化常数与\(\epsilon\)无关，所有未解决的尺度依赖性都隔离在最佳逼近误差中。然后，我们为具有三次反应的一维周期扩散方程构造了一个最小障碍见证。在单参数族\(v_c(x)=cx(1 - x)\)上，强残差的经验拉德马赫复杂度下限由\((\epsilon\sqrt{N})^{-1}\)的常数倍界定，而平方强残差损失的经验拉德马赫复杂度下限由\((\epsilon^2\sqrt{N})^{-1}\)的常数倍界定。这些是采样残差和损失类的与优化器无关的属性，而不是神经切线核条件语句。相应的变分能量复杂度在\(\epsilon\)上均匀地由\(N^{-1/2}\)的常数倍界定。张量积构造表明，在每个空间维度上相同的障碍率持续存在。数值评估分别给出了强残差、平方强损失和变分能量的拟合指数\(0.9971\)、\(1.9860\)和\(-0.0028\)。因此，区分微观系数会产生有限样本统计病态。变分公式消除了这种统计惩罚，但没有消除单独的多尺度逼近问题。
英文摘要
We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale \(\varepsilon\). Assuming a quantitative corrected \(H^1\)-estimate, a two-scale state class yields \[ \mathcal A_m^\varepsilon \le C\bigl(\varepsilon+Φ_{0,m_0}^2+Φ_{1,m_1}^2\bigr) \] in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining \(O(\varepsilon)\) approximation, state and flux feature errors, empirical sampling error, and an \(O(K^{-1})\) optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of \((\varepsilon\sqrt N)^{-1}\) and \((\varepsilon^2\sqrt N)^{-1}\), respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted \(\varepsilon\)- and \(N\)-scalings for nonlinear fluxes in \(d=1,2,3\), validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and \(H^1\) errors as the microscopic scale is refined.
Comments21 pages, 7 figures, 6 tables
