  We present a novel deep operator network (DeepONet) architecture for operator
learning, the ensemble DeepONet, that allows for enriching the trunk network of
a single DeepONet with multiple distinct trunk networks. This trunk enrichment
allows for greater expressivity and generalization capabilities over a range of
operator learning problems. We also present a spatial mixture-of-experts (MoE)
DeepONet trunk network architecture that utilizes a partition-of-unity (PoU)
approximation to promote spatial locality and model sparsity in the operator
learning problem. We first prove that both the ensemble and PoU-MoE DeepONets
are universal approximators. We then demonstrate that ensemble DeepONets
containing a trunk ensemble of a standard trunk, the PoU-MoE trunk, and/or a
proper orthogonal decomposition (POD) trunk can achieve 2-4x lower relative
$\ell_2$ errors than standard DeepONets and POD-DeepONets on both standard and
challenging new operator learning problems involving partial differential
equations (PDEs) in two and three dimensions. Our new PoU-MoE formulation
provides a natural way to incorporate spatial locality and model sparsity into
any neural network architecture, while our new ensemble DeepONet provides a
powerful and general framework for incorporating basis enrichment in scientific
machine learning architectures for operator learning.
