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

arXiv: https://arxiv.org/abs/2302.08144 · HTML: https://mlsanity.com/paper/2302.08144/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 10 references that are in the map.
- First posted on arXiv 16 Feb 2023.

## Built on by (most cited first, 7 of 7)

- [Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems](https://mlsanity.com/paper/2308.07051/) (2023)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Neural operators struggle to learn complex PDEs in pedestrian mobility: Hughes model case study](https://mlsanity.com/paper/2504.18267/) (2025)
- [Urban traffic analysis and forecasting through shared Koopman eigenmodes](https://mlsanity.com/paper/2409.04728/) (2024)
- [Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs](https://mlsanity.com/paper/2601.06388/) (2026)
- [Fractional is Better: Learnable Derivative Orders in Neural Operator Learning](https://mlsanity.com/paper/2610.20363/) (2026)

## Abstract (arXiv)

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 (π-FNO) to learn the weak solutions. We empirically quantify the generalization/out-of-sample error of the π-FNO solver as a function of input complexity, i.e., the distributions of initial and boundary conditions. Our testing results show that π-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.