PDEformer-2: A Versatile Foundation Model for Two-Dimensional Partial Differential Equations

TL;DR

PDEformer-2 is a versatile foundation model for two-dimensional PDEs that can generate accurate, mesh-free solutions at arbitrary coordinates, capable of zero-shot generalization and efficient adaptation to new PDEs.

Contribution

It introduces PDEformer-2, a flexible, pretrained foundation model that encodes diverse 2D PDEs and enables rapid, accurate solutions with zero-shot and few-shot capabilities.

Findings

Achieves accurate zero-shot PDE predictions similar to training data.

Demonstrates faster learning and smaller errors with limited samples.

Effective in inverse problems for PDE coefficient recovery.

Abstract

Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with adequate accuracy, and limitations of the traditional solvers and specialized neural operators motivate the development of foundation models for solving PDEs. This paper introduces PDEformer-2, a versatile foundation model for two-dimensional PDEs. Based on our previous one-dimensional PDEformer-1 model, PDEformer-2 receives the PDE form as network input via computational graph representation, which has the flexibility to encode most common PDEs. The mesh-free predicted solutions can be directly queried at arbitrary spatio-temporal coordinates. A large (40TB) diverse dataset is employed to pretrain the current model, making it capable of simultaneously addressing PDEs with different symbolic forms, domain shapes, boundary conditions, number of variables, and time-dependency. Accurate zero-shot prediction is allowed for PDEs that resemble the pretraining ones. When adapted to new unseen PDEs, PDEformer-2 demonstrates faster learning than many specialized models, and has smaller errors given limited (less than 100) samples. Additionally, PDEformer-2 can be employed in the inverse problems thanks to its fast and differentiable nature and produces reasonable results in our experiments to recover coefficient scalars and fields of a PDE.