  Neural Operator Networks (ONets) represent a novel advancement in machine
learning algorithms, offering a robust and generalizable alternative for
approximating partial differential equations (PDEs) solutions. Unlike
traditional Neural Networks (NN), which directly approximate functions, ONets
specialize in approximating mathematical operators, enhancing their efficacy in
addressing complex PDEs. In this work, we evaluate the capabilities of Deep
Operator Networks (DeepONets), an ONets implementation using a branch/trunk
architecture. Three test cases are studied: a system of ODEs, a general
diffusion system, and the convection/diffusion Burgers equation. It is
demonstrated that DeepONets can accurately learn the solution operators,
achieving prediction accuracy scores above 0.96 for the ODE and diffusion
problems over the observed domain while achieving zero shot (without
retraining) capability. More importantly, when evaluated on unseen scenarios
(zero shot feature), the trained models exhibit excellent generalization
ability. This underscores ONets vital niche for surrogate modeling and digital
twin development across physical systems. While convection-diffusion poses a
greater challenge, the results confirm the promise of ONets and motivate
further enhancements to the DeepONet algorithm. This work represents an
important step towards unlocking the potential of digital twins through robust
and generalizable surrogates.
