神经算子在核方法下实现最优收敛，理论严谨且可应用于偏微分方程求解。
Optimal Convergence Rates for Neural Operators
- 基于神经正切核框架分析两层神经算子的泛化性能
- 早期停止梯度下降达到极小极大最优收敛率
- 适用于需要高精度代理模型的科学计算场景
我们引入了两层神经算子的神经正切核（NTK）框架，并分析其泛化性质。对于早期停止的梯度下降（GD），我们推导出在再生核希尔伯特空间（RKHS）非参数回归框架下已知的极小极大最优收敛速率。我们给出了保证泛化所需的隐藏神经元数量和第二阶段样本数的上界。为验证我们的NTK框架，我们进一步证明：任何可由神经算子近似的算子，也可由RKHS中的算子近似。神经算子的关键应用之一是学习偏微分方程（PDE）解算子的代理映射。我们以标准泊松方程为例，通过模拟验证了理论结果。
原文摘要 · Abstract (English)
We introduce the neural tangent kernel (NTK) regime for two-layer neural operators and analyze their generalization properties. For early-stopped gradient descent (GD), we derive fast convergence rates that are known to be minimax optimal within the framework of non-parametric regression in reproducing kernel Hilbert spaces (RKHS). We provide bounds on the number of hidden neurons and the number of second-stage samples necessary for generalization. To justify our NTK regime, we additionally show that any operator approximable by a neural operator can also be approximated by an operator from the RKHS. A key application of neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider the standard Poisson equation to illustrate our theoretical findings with simulations.
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