  This paper introduces PDEformer-1, a versatile neural solver capable of
simultaneously addressing various partial differential equations (PDEs). With
the PDE represented as a computational graph, we facilitate the seamless
integration of symbolic and numeric information inherent in a PDE. A graph
Transformer and an implicit neural representation (INR) are employed
subsequently to generate mesh-free predicted solutions. We generated a dataset
with up to three million samples involving diverse one-dimensional PDEs to
pretrain our model. Compared with baseline models trained specifically on
benchmark datasets, our pretrained model achieves comparable accuracy via
zero-shot inference, and the advantage expands after finetuning. For PDEs new
or unseen in the pretraining stage, our model can adapt quickly by finetuning
on a relatively small set of examples from the target equation. Additionally,
PDEformer-1 demonstrates promising results in the inverse problem of PDE scalar
coefficient recovery and coefficient field recovery.
