Existing architectures for operator learning require that the number and locations of sensors (where the input functions are evaluated) remain the same across all training and test samples, significantly restricting the range of their applicability. We address this issue by proposing a novel operator learning framework, termed Variable-Input Deep Operator Network (VIDON), which allows for random sensors whose number and locations can vary across samples. VIDON is invariant to permutations of sensor locations and is proved to be universal in approximating a class of continuous operators. We also prove that VIDON can efficiently approximate operators arising in PDEs. Numerical experiments with a diverse set of PDEs are presented to illustrate the robust performance of VIDON in learning operators.

翻译：现有的操作者学习结构要求所有培训和测试样本的传感器数量和位置(在其中评价输入功能)保持不变,大大限制了其适用性,我们通过提出一个新的操作者学习框架(称为变数-投入深度操作者网络(VIDON))来解决这一问题,该框架允许随机传感器,其数量和位置可以因抽样而异。VIDON不易移动传感器位置,在类似连续操作者类别时被证明是普遍性的。我们还证明VIDON能够有效地接近PDEs中产生的操作者。用一套不同的PDEs的量化实验展示了VIDON在学习操作者的有力表现。