Variable-Input Deep Operator Networks
This addresses a key bottleneck in operator learning for scientific computing, enabling more flexible and practical applications in PDE modeling.
The paper tackles the limitation of fixed sensor locations in operator learning by proposing VIDON, a framework that handles variable numbers and positions of sensors across samples, achieving robust performance in learning operators from PDEs.
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.