用神经网络高效逼近分数阶抛物方程解，突破维度瓶颈。
Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations
- 引入各向异性谱Barron空间，分离时空频率正则性分析。
- 证明n^{-1/2}的两层网络逼近误差，在非周期激活下需多项式衰减条件。
- 首次实现时间全局一致的正则性分析，适合高维偏微分方程研究者。
我们建立了含低阶漂移与势项的分数阶抛物方程解的维度高效神经网络逼近理论。通过引入各向异性谱Barron空间，分别度量频率空间中的时间和空间正则性，首先发展了该类方程的维度无关最大正则性理论，利用维度无关乘法估计与连续性方法处理低阶项。关键技术创新在于将Vandermonde矩阵用于有限时间分数热半群的时间全局扩展，确保初始时刻足够光滑，从而借助各向异性Barron范数的全局时空傅里叶结构分析前向演化。我们还证明，对应的谱Barron正则性在时间上通常无法保持一致。最后，推导出非恒定周期激活函数下混合Sobolev范数中的n^{-1/2}两层逼近界，并在额外各向异性Barron正则性条件下，对满足多项式衰减条件的非周期激活函数也获得类似结果。
原文摘要 · Abstract (English)
We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。