We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of recovering the solution operator of hyperbolic PDEs is the presence of characteristics, along which the associated Green's function is discontinuous. Therefore, a central component of our algorithm is a rank detection scheme that identifies the approximate location of the characteristics. By combining the randomized singular value decomposition with an adaptive hierarchical partition of the domain, we construct an approximant to the solution operator using $O(\Psi_\epsilon^{-1}\epsilon^{-7}\log(\Xi_\epsilon^{-1}\epsilon^{-1}))$ input-output pairs with relative error $O(\Xi_\epsilon^{-1}\epsilon)$ in the operator norm as $\epsilon\to0$, with high probability. Here, $\Psi_\epsilon$ represents the existence of degenerate singular values of the solution operator, and $\Xi_\epsilon$ measures the quality of the training data. Our assumptions on the regularity of the coefficients of the hyperbolic PDE are relatively weak given that hyperbolic PDEs do not have the ``instantaneous smoothing effect'' of elliptic and parabolic PDEs, and our recovery rate improves as the regularity of the coefficients increases.

翻译：我们构建了第一个经过严格证明的概率算法，用于从输入-输出训练对中恢复双曲型偏微分方程（PDE）在两变量情况下的解算子。恢复双曲型PDE解算子的主要挑战在于特征线的存在，沿着这些特征线，相关的格林函数是不连续的。因此，我们算法的核心组成部分是一个秩检测方案，用于识别特征线的近似位置。通过将随机奇异值分解与域的自适应层次划分相结合，我们利用$O(\Psi_\epsilon^{-1}\epsilon^{-7}\log(\Xi_\epsilon^{-1}\epsilon^{-1}))$个输入-输出对构造了解算子的近似值，在算子范数下相对误差为$O(\Xi_\epsilon^{-1}\epsilon)$（当$\epsilon\to0$时），且具有高概率。这里，$\Psi_\epsilon$表示解算子退化奇异值的存在性，而$\Xi_\epsilon$衡量训练数据的质量。鉴于双曲型PDE缺乏椭圆型和抛物型PDE的“瞬时平滑效应”，我们对双曲型PDE系数正则性的假设相对较弱，并且恢复速率随着系数正则性的提高而提升。