Unlike traditional mesh-based approximations of differential operators, machine learning methods, which exploit the automatic differentiation of neural networks, have attracted increasing attention for their potential to mitigate stability issues encountered in the numerical simulation of hyperbolic conservation laws. However, solutions to hyperbolic problems are often piecewise smooth, rendering the differential form invalid along discontinuity interfaces and limiting the effectiveness of standard learning approaches. In this work, we propose lift-and-embed learning methods for solving scalar hyperbolic equations with discontinuous solutions, which consist of (i) embedding the Rankine-Hugoniot jump condition within a higher-dimensional space through the inclusion of an augmented variable in the solution ansatz; (ii) utilizing physics-informed neural networks to manage the increased dimensionality and to address both linear and quasi-linear problems within a unified learning framework; and (iii) projecting the trained network solution back onto the original lower-dimensional plane to obtain the approximate solution. Besides, the location of discontinuity can be parametrized as extra model parameters and inferred concurrently with the training of network solution. With collocation points sampled on piecewise surfaces rather than distributed over the entire lifted space, we conduct numerical experiments on various benchmark problems to demonstrate the capability of our methods in resolving discontinuous solutions without spurious numerical smearing and oscillations.

翻译：与传统的基于网格的微分算子逼近方法不同，利用神经网络自动微分功能的机器学习方法，因其在双曲守恒律数值模拟中缓解稳定性问题的潜力而日益受到关注。然而，双曲型问题的解通常是分段光滑的，这导致微分形式在间断界面处失效，从而限制了标准学习方法的有效性。本文提出了一种提升与嵌入学习方法，用于求解具有间断解的标量双曲型方程。该方法包括：（i）通过在解的假设形式中引入增广变量，将Rankine-Hugoniot跳跃条件嵌入到更高维的空间中；（ii）利用物理信息神经网络处理增加的维度，并在统一的学习框架内同时处理线性和拟线性问题；（iii）将训练好的网络解投影回原始的低维平面以获得近似解。此外，间断的位置可以作为额外的模型参数进行参数化，并在网络解的训练过程中同步推断。通过在分段曲面上而非整个提升空间内分布地采集配置点，我们对各种基准问题进行了数值实验，以证明我们的方法能够在没有虚假数值弥散和振荡的情况下解析间断解。