Multi-scale time-stepping of Partial Differential Equations with transformers

Developing fast surrogates for Partial Differential Equations (PDEs) will accelerate design and optimization in almost all scientific and engineering applications. Neural networks have been receiving ever-increasing attention and demonstrated remarkable success in computational modeling of PDEs, however; their prediction accuracy is not at the level of full deployment. In this work, we utilize the transformer architecture, the backbone of numerous state-of-the-art AI models, to learn the dynamics of physical systems as the mixing of spatial patterns learned by a convolutional autoencoder. Moreover, we incorporate the idea of multi-scale hierarchical time-stepping to increase the prediction speed and decrease accumulated error over time. Our model achieves similar or better results in predicting the time-evolution of Navier–Stokes equations compared to the powerful Fourier Neural Operator (FNO) and two transformer-based neural operators OFormer and Galerkin Transformer. The code and data are available on https: //github. com/BaratiLab/MSTPDE.