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.

翻译：基于数据驱动方法求解偏微分方程已日益普遍。算子学习范式的最新进展使得求解更广泛的偏微分方程相关问题成为可能。我们提出一种算子学习方法，可在无需任何时间离散化的情况下连续求解含时偏微分方程。该方法命名为DiTTO，灵感源自潜在扩散模型。尽管扩散模型通常用于生成式人工智能任务，但其时间条件机制对偏微分方程极具价值。我们将扩散启发框架与Transformer架构元素相结合以增强其能力。我们在多类多维偏微分方程上验证了新方法的有效性，包括一维Burgers方程、二维Navier-Stokes方程以及二维和三维声波方程。DiTTO在这些问题上取得了当前最优的精度结果。我们还提出一种利用扩散模型快速采样概念来提升DiTTO性能的方法。最后，我们证明DiTTO能够在时间维度上准确执行零样本超分辨率。