Transformer for Partial Differential Equations' Operator Learning
This work addresses the challenge of approximating PDE solutions for computational science and engineering, representing an incremental improvement by adapting transformer architectures to operator learning.
The authors tackled the problem of learning solution operators for partial differential equations by proposing an attention-based framework called Operator Transformer (OFormer), which is competitive on standard benchmarks and adaptable to randomly sampled inputs.
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.