  Data-driven learning of partial differential equations' solution operators
has recently emerged as a promising paradigm for approximating the underlying
solutions. The solution operators are usually parameterized by deep learning
models that are built upon problem-specific inductive biases. An example is a
convolutional or a graph neural network that exploits the local grid structure
where functions' values are sampled. The attention mechanism, on the other
hand, provides a flexible way to implicitly exploit the patterns within inputs,
and furthermore, relationship between arbitrary query locations and inputs. In
this work, we present an attention-based framework for data-driven operator
learning, which we term Operator Transformer (OFormer). Our framework is built
upon self-attention, cross-attention, and a set of point-wise multilayer
perceptrons (MLPs), and thus it makes few assumptions on the sampling pattern
of the input function or query locations. We show that the proposed framework
is competitive on standard benchmark problems and can flexibly be adapted to
randomly sampled input.
