Transolver: A Fast Transformer Solver for PDEs on General Geometries

变形金刚已经在各个领域赋能了许多里程碑，并且最近被应用于解决偏微分方程（PDEs）。然而，由于PDEs通常被离散成具有复杂几何形状的大规模网格，因此对于变形金刚来直接捕捉大量个体点中的复杂物理相关性是具有挑战性的。为了超越表面和笨重的网格，我们提出了基于一种更基础的思想的Transolver，即学习隐藏在离散几何形状背后的内在物理状态。具体而言，我们提出了一种新的物理注意力机制，以自适应地将离散域分割成一系列可学习的灵活形状的切片，其中物理状态相似的网格点将被归属于同一切片。通过计算从切片编码的物理感知标记的注意力，Transolver可以有效地捕捉复杂几何形状下的复杂物理相关性，这也赋予了求解器内生几何通用建模能力，并且可以以线性复杂度高效计算。Transolver在六个标准基准测试中实现了一致的最新技术水平，相对增益为22％，并且在大规模工业模拟中也表现出色，包括汽车和翼型设计。

Transolver: Learning to Solve Partial Differential Equations with Adaptive Domain Slicing and Physics-Attention

Transolver proposes a new Physics-Attention to adaptively split the discretized domain into a series of learnable slices of flexible shapes, where mesh points under similar physical states will be ascribed to the same slice. By calculating attention to physics-aware tokens encoded from slices, Transovler can effectively capture intricate physical correlations under complex geometrics, which also empowers the solver with endogenetic geometry-general modeling capacity and can be efficiently computed in linear complexity.

Transolver achieves consistent state-of-the-art with 22% relative gain across six standard benchmarks and also excels in large-scale industrial simulations, including car and airfoil designs.

Recent related works include NeuralPDE, Deep BSDE Solver, and PINN.