用神经网络同时逼近微分方程解与边界条件，提升高频率问题求解精度。
Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
- 基于流形上的谱边界损失，避免传统边界处理的复杂构造。
- 在球面半区和半环面测试中，精度提升1~2个数量级。
- 适合需高精度边界满足的偏微分方程求解任务。
本文提出在紧致黎曼流形上同时进行Sobolev逼近与椭圆型边值问题求解的卷积神经网络方法。证明了单通道与多通道CNN的逼近误差估计，其收敛率由内在维数与光滑性间隙决定。为匹配椭圆稳定性，设计物理信息神经网络框架，采用谱边界损失：边界残差展开为边界拉普拉斯-贝尔特拉米本征模，以Sobolev迹权重惩罚，对应2s阶椭圆问题的自然迹范数\(\mathcal H^{2s-1/2}(oundary\mathcal M^d)\)。该方法避免精确边界强制所需的平滑辅助构造及奇异的Sobolev-Slobodeckij双重积分，支持基于FFT或预计算谱实现。推导出误差分解，分离逼近、泛化与谱截断误差，表明所提损失与局部快速率泛化分析一致。数值实验在上半球面与上半环面验证，相比标准PINNs显著提升精度、收敛性与稳定性，对高频边界数据实现1~2个数量级的性能增益。
原文摘要 · Abstract (English)
This paper develops convolutional neural network (CNN) methods for simultaneous Sobolev approximation and elliptic boundary value problems on compact Riemannian manifolds. We prove approximation estimates for single- and multichannel CNNs, with rates governed by the intrinsic dimension and the smoothness gap. Motivated by elliptic stability, we propose a physics-informed CNN framework with a spectral boundary loss. The boundary residual is expanded in boundary Laplace--Beltrami eigenmodes and penalized by Sobolev trace weights, matching the natural \(\mathcal H^{2s-1/2}(\partial\mathcal M^d)\) trace norm for \(2s\)-order elliptic problems. This avoids smooth auxiliary constructions for exact boundary enforcement and singular Sobolev--Slobodeckij double integrals, while allowing FFT-based or precomputed spectral implementations. We also derive an error decomposition separating approximation, generalization, and spectral truncation errors, showing that the proposed loss is aligned with localized fast-rate generalization analysis. Numerical experiments on the upper hemisphere and upper half-torus demonstrate improved accuracy, convergence, and stability over standard PINNs, with one to two orders of magnitude gains for high-frequency boundary data.
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