Operator learning for hyperbolic partial differential equations
Christopher Wang Alex Townsend
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.