AI 中文总结
针对非线性非均匀椭圆问题，提出切空间多尺度流形方法，通过非线性重构粗状态表示细尺度解，建立粗方程，还有网络插值变体，记录相关特性，有条件控制局部化缺陷并证明其衰减。
AI 中文摘要
我们为非线性非均匀椭圆问题引入一种切空间多尺度流形方法。该方法通过对粗状态进行非线性重构来表示细尺度解。理想重构通过约束变分问题消除细尺度，可计算重构通过与单位分解混合的局部非线性补丁求解来近似此映射。由于近似集是一个非线性流形，粗方程使用切多尺度测试函数来建立。我们还制定了一种网络插值变体，其中只有单位分解混合使用的受限补丁输出及其切向作用由局部学习映射近似。对于非均匀单调非线性扩散，我们记录了结构单调性、可微性、补丁映射正则性和条件扰动估计，这些将几何稳定性机制与局部化、残差和可选学习缺陷区分开来。关于局部化缺陷衰减的严格先验理论以及由此产生的粗网格尺寸收敛率将在单独分析中给出；在此我们有条件地控制这些缺陷并通过数值证明其衰减。
英文摘要
We introduce a tangent-space multiscale manifold method for nonlinear heterogeneous elliptic problems. The method represents the fine-scale solution by a nonlinear reconstruction of a coarse state. The ideal reconstruction eliminates fine scales through a constrained variational problem, and the computable reconstruction approximates this map by localized nonlinear patch solves blended with a partition of unity. Because the approximation set is a nonlinear manifold, the coarse equation is posed with tangent multiscale test functions. We also formulate a network-interpolated variant in which only the restricted patch outputs used by the partition-of-unity blend, together with their tangent actions, are approximated by local learned maps. For heterogeneous monotone nonlinear diffusion, we record the structural monotonicity, differentiability, patch-map regularity, and conditional perturbation estimates that separate the geometric stability mechanism from localization, residual, and optional learning defects. A rigorous a priori theory for the decay of the localization defect, and the resulting convergence rates in the coarse mesh size, is deferred to a separate analysis; here these defects are controlled conditionally and their decay is demonstrated numerically.
