提出可证明误差的神经网络方法，解决多尺度非线性椭圆方程求解难题
Non-Asymptotic Variational Learning for Monotone Nonlinear Multiscale Elliptic Equations: Scale-Robust Primal-Dual Bounds and Strong-Form Statistical Ill-Conditioning
- 构建基于物理约束的变分学习框架，误差分解为近似、采样与优化项
- 在任意维度下实现O(ε)精度，且所有常数与微观尺度ε无关
- 首次给出强残差类的统计病态下界，适合研究多尺度模型泛化性
我们发展了用于均匀单调非线性多尺度椭圆方程的非渐近变分物理信息近似理论，涵盖逼近、采样与有限迭代优化。对于边界相容的神经特征类，总体误差分解为近似、经验积分与投影梯度项，所有非近似常数在微观尺度ε上一致。假设量化修正的H¹估计，双尺度状态类满足 \\[ \mathcal A_m^\varepsilon \le C\bigl(\varepsilon+Φ_{0,m_0}^2+Φ_{1,m_1}^2\bigr) \\[ 在任意维度成立。进一步引入凸对偶物理损失，其期望值为状态误差的可计算上界。在通量校正器正则性条件下，散度相容的双尺度通量类给出结合O(ε)近似、状态与通量特征误差、采样误差及O(K⁻¹)优化项的认证状态-通量界。相反，对于满足自然非退化条件的通用周期性非线性通量，强残差与平方残差类的经验Rademacher复杂度分别被下界控制于常数倍的(ε√N)⁻¹与(ε²√N)⁻¹。这些与优化器无关的下界在所有空间维数中成立。数值实验验证了d=1,2,3时非线性通量的ε和N标度预测，验证了所有计算的对偶证书，并表明校正器增强类在微观尺度细化时显著降低能量与H¹误差。
原文摘要 · Abstract (English)
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.
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