  The Deep operator network (DeepONet) is a powerful yet simple neural operator
architecture that utilizes two deep neural networks to learn mappings between
infinite-dimensional function spaces. This architecture is highly flexible,
allowing the evaluation of the solution field at any location within the
desired domain. However, it imposes a strict constraint on the input space,
requiring all input functions to be discretized at the same locations; this
limits its practical applications. In this work, we introduce a Resolution
Independent Neural Operator (RINO) that provides a framework to make DeepONet
resolution-independent, enabling it to handle input functions that are
arbitrarily, but sufficiently finely, discretized. To this end, we propose a
dictionary learning algorithm to adaptively learn a set of appropriate
continuous basis functions, parameterized as implicit neural representations
(INRs), from the input data. These basis functions are then used to project
arbitrary input function data as a point cloud onto an embedding space (i.e., a
vector space of finite dimensions) with dimensionality equal to the dictionary
size, which can be directly used by DeepONet without any architectural changes.
In particular, we utilize sinusoidal representation networks (SIRENs) as our
trainable INR basis functions. We demonstrate the robustness and applicability
of RINO in handling arbitrarily (but sufficiently richly) sampled input
functions during both training and inference through several numerical
examples.
