Solving partial differential equations (PDEs) using a data-driven approach has become increasingly common. The recent development of the operator learning paradigm has enabled the solution of a broader range of PDE-related problems. We propose an operator learning method to solve time-dependent PDEs continuously in time without needing any temporal discretization. The proposed approach, named DiTTO, is inspired by latent diffusion models. While diffusion models are usually used in generative artificial intelligence tasks, their time-conditioning mechanism is extremely useful for PDEs. The diffusion-inspired framework is combined with elements from the Transformer architecture to improve its capabilities. We demonstrate the effectiveness of the new approach on a wide variety of PDEs in multiple dimensions, namely the 1-D Burgers' equation, 2-D Navier-Stokes equations, and the acoustic wave equation in 2-D and 3-D. DiTTO achieves state-of-the-art results in terms of accuracy for these problems. We also present a method to improve the performance of DiTTO by using fast sampling concepts from diffusion models. Finally, we show that DiTTO can accurately perform zero-shot super-resolution in time.

我们提出了一种名为DiTTO的算子学习方法来连续地解决时间相关的偏微分方程，该方法通过将受扩散模型启发的框架与Transformer架构相结合，实现了在多个维度上的各种PDEs的准确解决，并通过使用扩散模型中的快速采样概念进一步提高了性能，并展示了DiTTO可以在时间上精确执行零短片超分辨率。

DiTTO: 扩散启发的时态变换器操作符