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