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.

翻译：神经算子已广泛应用于科学领域，如求解参数化偏微分方程、带控制的动力系统及反问题。然而，当处理具有异质特性的输入函数时面临挑战，这需要多传感器处理正则性最低的函数。为应对此问题，离散化不变的神经算子已被采用，允许对不同传感器位置采样的多样化输入函数进行建模。但现有框架仍要求所有函数具有相同数量的传感器。本研究提出一种新颖的分布式方法，进一步放宽离散化要求并解决异构数据集难题。该方法通过划分输入函数空间，利用独立的神经网络处理各输入函数，并采用集中式神经网络处理所有输出函数的共享信息。这种分布式方法减少了梯度下降反向传播步骤，在保持精度的同时提升了效率。我们证明了相应神经网络是连续非线性算子的通用逼近器，并通过四个数值算例验证了其性能。