Operator learning has historically struggled to deal with PDEs with advection. We show that recovering the solution operator for advective PDEs is mathematically possible, but difficult. Specifically, we construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic PDE in two variables, using input-output training pairs. The primary challenge 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. We will also discuss some theoretical challenges and potential steps to address them, as well as potential lower bounds on our convergence rate.

Bio: Christopher Wang is a PhD candidate at Cornell University being advised by Alex Townsend. His primary research interests are in randomized numerical linear algebra and numerical aspects of PDE theory.