Lift-and-Embed Learning Methods for Solving Scalar Hyperbolic Equations with Discontinuous Solutions
Qi Sun, Zhenjiang Liu, Lili Ju, Xuejun Xu
TL;DR
This work tackles the challenge of resolving discontinuities in scalar hyperbolic conservation laws using a mesh-free, neural-network–based approach. It introduces a lift-and-embed framework that embeds the Rankine–Hugoniot jump condition in a one-order higher-dimensional space via an augmented variable, enabling a smooth representation of discontinuities and accurate reconstruction on the physical domain. The method supports both known and unknown shock locations, using a residual-minimization training scheme on collocation points placed on piecewise lifted surfaces and, when needed, jointly infers shock paths through backpropagation. Across linear and nonlinear 1D problems, shocks and rarefactions, and a 2D Burgers’ test, the approach achieves sharp interfaces without spurious smearing or oscillations, illustrating a unified, scalable framework with potential for extension to hyperbolic systems and operator learning.
Abstract
Deep learning methods, which exploit auto-differentiation to compute derivatives without dispersion or dissipation errors, have recently emerged as a compelling alternative to classical mesh-based numerical schemes for solving hyperbolic conservation laws. However, solutions to hyperbolic problems are often piecewise smooth, posing challenges for training of neural networks to capture solution discontinuities and jumps across interfaces. In this paper, we propose a novel lift-and-embed learning method to effectively resolve these challenges. The proposed method comprises three innovative components: (i) embedding the Rankine-Hugoniot condition within a one-order higher-dimensional space by including an augmented variable; (ii) utilizing neural networks to handle the increased dimensionality and address both linear and nonlinear problems within a unified mesh-free learning framework; and (iii) projecting the trained model back onto the original physical domain to obtain the approximate solution. Notably, the location of discontinuities also can be treated as trainable parameters in our method and inferred concurrently with the training of neural network solutions. With collocation points sampled only on piecewise surfaces rather than fulfilling the whole lifted space, we demonstrate through extensive numerical experiments that our method can efficiently and accurately solve scalar hyperbolic equations with discontinuous solutions without spurious smearing or oscillations.