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：我们将视觉变换器与算子学习相结合，以求解由偏微分方程描述的各类逆问题。我们的方法名为ViTO，融合了基于U-Net的架构与视觉变换器。我们将ViTO应用于求解复杂度递增的偏微分方程逆问题，具体涉及波动方程、纳维-斯托克斯方程及达西方程。我们重点研究了更具挑战性的超分辨率情形，其中逆问题的输入数据集分辨率显著低于输出。在精度方面，我们获得的结果可与领先的算子网络基准相媲美甚至更优。此外，ViTO架构的可训练参数量较少（不足领先对手的10%），从而使各测试案例的平均性能提升超过5倍。