We present a generalized version of the discretization-invariant neural operator and prove that the network is a universal approximation in the operator sense. Moreover, by incorporating additional terms in the architecture, we establish a connection between this discretization-invariant neural operator network and those discussed before. The discretization-invariance property of the operator network implies that different input functions can be sampled using various sensor locations within the same training and testing phases. Additionally, since the network learns a ``basis'' for the input and output function spaces, our approach enables the evaluation of input functions on different discretizations. To evaluate the performance of the proposed discretization-invariant neural operator, we focus on challenging examples from multiscale partial differential equations. Our experimental results indicate that the method achieves lower prediction errors compared to previous networks and benefits from its discretization-invariant property.

翻译：我们提出了一种广义版本的离散化不变神经算子，并证明了该网络在算子意义上具有通用逼近性。此外，通过在架构中引入额外项，我们建立了该离散化不变神经算子网络与先前讨论的网络之间的联系。算子网络的离散化不变性意味着，在同一训练和测试阶段，不同的输入函数可以使用不同的传感器位置进行采样。同时，由于该网络学习了输入和输出函数空间的"基"，我们的方法支持对输入函数在不同离散化网格上进行评估。为了评估所提出的离散化不变神经算子的性能，我们重点关注来自多尺度偏微分方程的具有挑战性的算例。实验结果表明，与现有网络相比，该方法实现了更低的预测误差，并得益于其离散化不变特性。