In recent years, Solving partial differential equations has shifted the focus of traditional neural network studies from finite-dimensional Euclidean spaces to generalized functional spaces in research. A novel methodology is to learn an operator as a means of approximating the mapping between outputs. Currently, researchers have proposed a variety of operator architectures. Nevertheless, the majority of these architectures adopt an iterative update architecture, whereby a single operator is learned from the same function space. In practical physical science problems, the numerical solutions of partial differential equations are complex, and a serial single operator is unable to accurately approximate the intricate mapping between input and output. So, We propose a deep parallel operator model (DPNO) for efficiently and accurately solving partial differential equations. DPNO employs convolutional neural networks to extract local features and map data into distinct latent spaces. Designing a parallel block of double Fourier neural operators to solve the iterative error problem. DPNO approximates complex mappings between inputs and outputs by learning multiple operators in different potential spaces in parallel blocks. DPNO achieved the best performance on five of them, with an average improvement of 10.5\%, and ranked second on one dataset.

本研究针对传统神经网络方法在解决偏微分方程时的局限性，提出了一种新的深度并行算子模型（DPNO），有效解决了单一算子在复杂映射中无法准确逼近的问题。DPNO通过并行学习多个算子，显著提高了对偏微分方程解的近似精度，平均提升了10.5%。

使用深度并行神经算子的偏微分方程学习