Neural operators have been applied in various scientific fields, such as solving parametric partial differential equations, dynamical systems with control, and inverse problems. However, challenges arise when dealing with input functions that exhibit heterogeneous properties, requiring multiple sensors to handle functions with minimal regularity. To address this issue, discretization-invariant neural operators have been used, allowing the sampling of diverse input functions with different sensor locations. However, existing frameworks still require an equal number of sensors for all functions. In our study, we propose a novel distributed approach to further relax the discretization requirements and solve the heterogeneous dataset challenges. Our method involves partitioning the input function space and processing individual input functions using independent and separate neural networks. A centralized neural network is used to handle shared information across all output functions. This distributed methodology reduces the number of gradient descent back-propagation steps, improving efficiency while maintaining accuracy. We demonstrate that the corresponding neural network is a universal approximator of continuous nonlinear operators and present four numerical examples to validate its performance.

我们提出了一种新的分布式方法来放宽离散化要求，解决异构数据集挑战。我们的方法涉及将输入函数空间划分，并使用独立和分开的神经网络处理单个输入函数。通过使用一个集中的神经网络处理所有输出函数中的共享信息，这种分布式方法减少了梯度下降反向传播步骤的数量，提高了效率并保持了准确性。我们证明了相应的神经网络是连续非线性算子的通用逼近器，并提出了四个数值示例来验证其性能。

D2NO: 处理分布式深度神经算子中异构输入函数空间的高效方法