## Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors

### Bilal Thonnam Thodi · Sai Venkata Ramana Ambadipudi · Saif Eddin Jabari

[ Abstract ]
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Abstract: 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 the 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.

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