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

TL;DR

The paper examines neural operators as mesh-independent surrogates for the Hughes pedestrian-model, a coupled conservation law and eikonal equation that forms nonlinear hyperbolic PDEs with shocks. It systematically tests three architectures—Fourier Neural Operator, Wavelet Neural Operator, and Multiwavelet Neural Operator—on datasets generated by two robust PDE solvers (Godunov and wave-front tracking) under varied initial and boundary conditions. Results show that while NOs can handle simple cases, they fail to preserve sharp discontinuities in complex, transport-dominated regimes, exhibiting smoothing behavior and diffusion-like errors that degrade shock visibility and transport dynamics; performance degrades with increasing discontinuities and boundary-driven effects. The findings underscore the need for shock-preserving, boundary-aware neural operator designs or training strategies to reliably model real-world crowd dynamics and similar hyperbolic systems, with practical implications for traffic and evacuation applications.

Abstract

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.

Figures (15)

• Figure 1: Layer-wise architecture of Neural Operator.
• Figure 2: Illustration of Hughes pedestrian dynamics* *High discretization used here for better visualization
• Figure 3: Example data for initial conditions with multiple discontinuities
• Figure 4: Example data for Godunov and WFT scheme, with same initial condition* *High discretization used here for better visualization
• Figure 5: Training data for initial value problem