We combine vision transformers with operator learning to solve diverse inverse problems described by partial differential equations (PDEs). Our approach, named ViTO, combines a U-Net based architecture with a vision transformer. We apply ViTO to solve inverse PDE problems of increasing complexity, namely for the wave equation, the Navier-Stokes equations and the Darcy equation. We focus on the more challenging case of super-resolution, where the input dataset for the inverse problem is at a significantly coarser resolution than the output. The results we obtain are comparable or exceed the leading operator network benchmarks in terms of accuracy. Furthermore, ViTO`s architecture has a small number of trainable parameters (less than 10% of the leading competitor), resulting in a performance speed-up of over 5x when averaged over the various test cases.

翻译：我们将视觉Transformer与运算器学习相结合，以解决由偏微分方程（PDE）描述的各种反问题。我们的方法名为ViTO，结合了基于U-Net的架构和视觉Transformer。我们将ViTO应用于解决日益复杂的反PDE问题，分别为波动方程、Navier-Stokes方程和Darcy方程。我们专注于更具挑战性的超分辨率情况，其中反问题的输入数据集比输出数据集粗糙得多。我们得到的结果在准确性上已经可以与领先的运算网络基准相媲美或超过。此外，ViTO的架构具有较少的可训练参数（不到领先竞争对手的10%），导致各种测试中平均表现速度提高了5倍以上。