证明神经算子可高效逼近非线性抛物方程解算子，理论更坚实。
Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations
- 用杜阿梅尔原理将偏微分方程转为积分方程，结合皮卡迭代思想分析
- 给出解算子的定量逼近误差上界，模型复杂度不指数增长
- 方法可推广至纳维-斯托克斯等可用皮卡迭代求解的方程
神经算子作为通用算子逼近器，在本文中我们推导了非线性抛物型偏微分方程（PDEs）解算子的逼近速率，为非线性PDE解算子的定量逼近定理提供了支持。结果表明，神经算子可在不引起模型复杂度指数增长的前提下，高效逼近这些解算子，从而强化了神经算子的理论基础。证明中的关键洞见是通过杜阿梅尔原理将PDE转化为对应的积分方程，并利用神经算子与皮卡迭代（Picard's iteration）之间的相似性。该方法具有潜在的普适性，可推广至一系列可用皮卡迭代求解的方程，包括纳维-斯托克斯方程、非线性薛定谔方程和非线性波动方程。
原文摘要 · Abstract (English)
Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard's iteration, a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a range of other equations, including the Navier-Stokes equation, nonlinear Schrödinger equations and nonlinear wave equations, which can be solved by Picard's iteration.
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