PDEformer-1: A Foundation Model for One-Dimensional Partial Differential Equations

TL;DR

PDEformer-1 is a versatile foundation model for one-dimensional PDEs that leverages graph transformers and implicit neural representations to solve, adapt, and perform inverse tasks with high efficiency and accuracy.

Contribution

The paper introduces PDEformer-1, a pretrained neural model capable of zero-shot PDE solving, quick adaptation to new PDEs, and inverse problem solving, integrating symbolic and numeric PDE information.

Findings

Achieves comparable accuracy to specialized models in zero-shot inference.

Adapts quickly to unseen PDEs through fine-tuning with few samples.

Performs well in inverse problems like coefficient recovery.

Abstract

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.