Operator learning for hyperbolic partial differential equations

2023-12-29

arXiv.org

AI Summary

This work addresses the problem of learning the solution operator of bivariate hyperbolic partial differential equations (PDEs) from input–output training pairs. The core challenge lies in the discontinuity of the Green’s function along characteristic curves, which invalidates conventional learning methods relying on smoothness assumptions typical of elliptic or parabolic PDEs. To overcome this, we propose the first theoretically grounded stochastic learning framework: it precisely locates characteristics via rank detection and constructs a characteristic-aware adaptive hierarchical randomized singular value decomposition (SVD) algorithm—thereby circumventing reliance on PDE solution smoothness. Under mild coefficient regularity, our method achieves an $O(Xi_varepsilon^{-1}varepsilon)$ relative error (in operator norm) with high probability, using only $O(Psi_varepsilon^{-1}varepsilon^{-7}log(Xi_varepsilon^{-1}varepsilon^{-1}))$ samples. This constitutes the first rigorous probabilistic learning result for hyperbolic PDE solution operators.

Abstract

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 $epsilon o0$, 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.

Problem

• Learning solution operators for hyperbolic PDEs
• Overcoming discontinuous Green's function challenges
• Developing probabilistic algorithm with adaptive partitioning

Innovation

• Probabilistic algorithm for hyperbolic PDE operator learning
• Rank detection scheme to identify discontinuous characteristics
• Randomized SVD with adaptive hierarchical domain partition