分数阶抛物型偏微分方程解的神经网络逼近
Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation
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中文总结 AI 辅助
该研究针对带低阶项的分数阶抛物型方程，引入各向异性谱Barron空间建立维度无关极大正则性理论，结合范德蒙德矩阵推导两层神经网络逼近界，为分数阶偏微分方程解的神经网络逼近提供了维度高效的理论支撑。
中文摘要 AI 辅助
我们针对带有低阶漂移项和势项的分数阶抛物型方程，建立了维度高效的神经网络逼近理论。通过引入各向异性谱Barron空间（在频率空间中分别度量时间和空间正则性），我们首先利用维度无关的乘法估计和连续性方法，将低阶项纳入考量，为这些方程发展出维度无关的极大正则性理论。一项关键技术创新是将范德蒙德矩阵应用于有限时间分数阶热半群的全局时间延拓，该半群在初始时刻具有足够正则性，从而能够通过各向异性Barron范数的全局时空傅里叶结构分析正向时间演化。我们还证明，谱Barron正则性的一致时间估计通常无法成立。最后，我们针对非常数周期激活函数，推导了混合索伯列夫范数下的n^{-1/2}两层逼近界；在额外的各向异性Barron正则性条件下，对于满足多项式衰减条件的非周期激活函数，也推导了相应的逼近界。
英文摘要
We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space--time Fourier $L^1$ norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and $γ$ derivatives in space, where $γ>0$ is the order of the fractional Laplacian. The evolution is defined only for $t\geq0$, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at $t=0$, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields $n^{-1/2}$ approximation rates for the solution in mixed space--time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-$n$ network for which the sum of the squared mixed-Sobolev solution error, the squared $L^2$-norm of the residual for the whole-space fractional equation, and the squared initial-data error is $O(n^{-1})$.
发表机构
- School of Mathematics, Korea Institute for Advanced Study（韩国高等研究院数学学院）
- Center for Artificial Intelligence and Natural Sciences, Korea Institute for Advanced Study（韩国高等研究院人工智能与自然科学中心）
- Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
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