We study learning weak solutions to nonlinear hyperbolic partial differential equations (H-PDE), which have been difficult to learn due to discontinuities in their solutions. We use a physics-informed variant of the Fourier Neural Operator ($\pi$-FNO) to learn the weak solutions. We empirically quantify the generalization/out-of-sample error of the $\pi$-FNO solver as a function of input complexity, i.e., the distributions of initial and boundary conditions. Our testing results show that $\pi$-FNO generalizes well to unseen initial and boundary conditions. We find that the generalization error grows linearly with input complexity. Further, adding a physics-informed regularizer improved the prediction of discontinuities in the solution. We use the Lighthill-Witham-Richards (LWR) traffic flow model as a guiding example to illustrate the results.

翻译：我们研究了学习非线性双曲型偏微分方程（H-PDE）弱解的方法，此类方程因解存在间断而难以学习。我们采用傅里叶神经算子的物理信息变体（π-FNO）来学习弱解。通过实验定量分析了π-FNO求解器作为输入复杂度（即初始条件和边界条件的分布）函数的泛化/样本外误差。测试结果表明，π-FNO能很好地泛化到未见过的初始条件和边界条件。我们发现泛化误差随输入复杂度线性增长。此外，加入物理信息正则化器可改善对解中间断点的预测。我们以Lighthill-Witham-Richards（LWR）交通流模型作为引导性示例来阐释相关结果。