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 Schr\"odinger equations and nonlinear wave equations, which can be solved by Picard's iteration.

翻译：神经算子作为一般连续算子的通用逼近器。本文推导了非线性抛物型偏微分方程解算子的逼近速率，为非线性偏微分方程解算子的定量逼近定理做出了贡献。我们的结果表明，神经算子能够高效地逼近这些解算子，且模型复杂度不会呈指数级增长，从而强化了神经算子的理论基础。我们证明中的一个关键见解是通过Duahamel原理将偏微分方程转化为相应的积分方程，并利用神经算子与Picard迭代（一种求解偏微分方程的经典算法）之间的相似性。该方法可能超越抛物型偏微分方程，推广至一系列其他方程，包括Navier-Stokes方程、非线性Schrödinger方程和非线性波动方程，这些方程均可通过Picard迭代求解。