Operator learning for hyperbolic partial differential equations
This addresses a fundamental challenge in operator learning for hyperbolic PDEs, which lack smoothing effects, offering a novel approach for applications in physics and engineering.
The paper tackles the problem of recovering the solution operator for hyperbolic PDEs from input-output pairs, presenting the first rigorously justified probabilistic algorithm that achieves a relative error of O(Ξ_ε^{-1}ε) in operator norm using O(Ψ_ε^{-1}ε^{-7}log(Ξ_ε^{-1}ε^{-1})) training pairs with high probability.
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 $O(Ψ_ε^{-1}ε^{-7}\log(Ξ_ε^{-1}ε^{-1}))$ input-output pairs with relative error $O(Ξ_ε^{-1}ε)$ in the operator norm as $ε\to0$, with high probability. Here, $Ψ_ε$ represents the existence of degenerate singular values of the solution operator, and $Ξ_ε$ 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.