Variable-Input Deep Operator Networks

About

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.

Michael Prasthofer, Tim De Ryck, Siddhartha Mishra · 2022

Related benchmarks

Task	Dataset	Result
Operator learning	Derivative Fixed	Relative L2 Error2.91	6
Operator learning	Integral Fixed	Relative L2 Error2.07	6
Operator learning	Elastic Plate Fixed	Relative L2 Error5.24	6
Operator learning	Darcy 1D Fixed	Relative L2 Error (Test)991	6
Diffraction	Point-cloud Diffraction	Relative L2 Error1.01	5
Operator learning	Darcy 1D Drop-off	Relative L2 Error (Test)1.05	5
Optimal Transport	Point-cloud Optimal Transport	Relative L2 Error0.0541	5
Heat conduction	Point-cloud Heat, M=30	Relative L2 Error2.2	5
Operator learning	Integral Variable	Relative L2 Error (Test)9.71	5
Advection Diffusion	Point-cloud Advection Diffusion	Relative L2 Error8.23	5