# Neural operators struggle to learn complex PDEs in pedestrian mobility: Hughes model case study

arXiv: https://arxiv.org/abs/2504.18267 · HTML: https://mlsanity.com/paper/2504.18267/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 28 references that are in the map.
- First posted on arXiv 25 Apr 2025.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [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)
- [Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations](https://mlsanity.com/paper/2201.11967/) (2022)
- [Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors](https://mlsanity.com/paper/2302.08144/) (2023)

## Built on by (most cited first, 5 of 5)

- [Bubbleformer: Forecasting Boiling with Transformers](https://mlsanity.com/paper/2507.21244/) (2025)
- [Efficient Dilated Squeeze and Excitation Neural Operator for Differential Equations](https://mlsanity.com/paper/2601.17407/) (2026)
- [Monte Carlo-Type Neural Operator for Differential Equations](https://mlsanity.com/paper/2510.05620/) (2025)
- [Deep Operator Networks for Surrogate Modeling of Cyclic Adsorption Processes with Varying Initial Conditions](https://mlsanity.com/paper/2601.09491/) (2026)
- [Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions](https://mlsanity.com/paper/2605.19823/) (2026)

## Abstract (arXiv)

This paper investigates the limitations of neural operators in learning solutions for a Hughes model, a first-order hyperbolic conservation law system for crowd dynamics. The model couples a Fokker-Planck equation representing pedestrian density with a Hamilton-Jacobi-type (eikonal) equation. This Hughes model belongs to the class of nonlinear hyperbolic systems that often exhibit complex solution structures, including shocks and discontinuities. In this study, we assess the performance of three state-of-the-art neural operators (Fourier Neural Operator, Wavelet Neural Operator, and Multiwavelet Neural Operator) in various challenging scenarios. Specifically, we consider (1) discontinuous and Gaussian initial conditions and (2) diverse boundary conditions, while also examining the impact of different numerical schemes. Our results show that these neural operators perform well in easy scenarios with fewer discontinuities in the initial condition, yet they struggle in complex scenarios with multiple initial discontinuities and dynamic boundary conditions, even when trained specifically on such complex samples. The predicted solutions often appear smoother, resulting in a reduction in total variation and a loss of important physical features. This smoothing behavior is similar to issues discussed by Daganzo (1995), where models that introduce artificial diffusion were shown to miss essential features such as shock waves in hyperbolic systems. These results suggest that current neural operator architectures may introduce unintended regularization effects that limit their ability to capture transport dynamics governed by discontinuities. They also raise concerns about generalizing these methods to traffic applications where shock preservation is essential.