Physics informed token transformer for solving partial differential equations

Abstract

Solving partial differential equations (PDEs) is the core of many fields of science and engineering. While classical approaches are often prohibitively slow, machine learning models often fail to incorporate complete system information. Over the past few years, transformers have had a significant impact on the field of Artificial Intelligence and have seen increased usage in PDE applications. However, despite their success, transformers currently lack integration with physics and reasoning. This study aims to address this issue by introducing Physics Informed Token Transformer (PITT). The purpose of PITT is to incorporate the knowledge of physics by embedding PDEs into the learning process. PITT uses an equation tokenization method to learn an analytically-driven numerical update operator. By tokenizing PDEs and embedding partial derivatives, the transformer models become aware of the underlying knowledge behind physical processes. To demonstrate this, PITT is tested on challenging 1D and 2D PDE operator learning tasks. The results show that PITT outperforms popular neural operator models and has the ability to extract physically relevant information from governing equations.