# Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems

arXiv: https://arxiv.org/abs/2308.07051 · HTML: https://mlsanity.com/paper/2308.07051/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 68 references that are in the map.
- First posted on arXiv 14 Aug 2023.

## Builds on (most cited first, 28 of 28)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [A Physics-Informed Deep Learning Paradigm for Car-Following Models](https://mlsanity.com/paper/2012.13376/) (2020)
- [A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation](https://mlsanity.com/paper/2106.03142/) (2021)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications](https://mlsanity.com/paper/2002.02374/) (2020)
- [A Survey on Autonomous Vehicle Control in the Era of Mixed-Autonomy: From Physics-Based to AI-Guided Driving Policy Learning](https://mlsanity.com/paper/2007.05156/) (2020)
- [Deep Operator Learning Lessens the Curse of Dimensionality for PDEs](https://mlsanity.com/paper/2301.12227/) (2023)
- [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models](https://mlsanity.com/paper/2302.12337/) (2023)
- [Incorporating Kinematic Wave Theory into a Deep Learning Method for High-Resolution Traffic Speed Estimation](https://mlsanity.com/paper/2102.02906/) (2021)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [Nonlinear Traffic Prediction as a Matrix Completion Problem with Ensemble Learning](https://mlsanity.com/paper/2001.02492/) (2020)
- [Learning-based Traffic State Reconstruction using Probe Vehicles](https://mlsanity.com/paper/2011.05031/) (2020)
- [Short-term traffic prediction using physics-aware neural networks](https://mlsanity.com/paper/2109.10253/) (2021)
- [A Generative Car-following Model Conditioned On Driving Styles](https://mlsanity.com/paper/2112.05399/) (2021)
- [Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors](https://mlsanity.com/paper/2302.08144/) (2023)
- [Traffic state estimation using stochastic Lagrangian dynamics](https://mlsanity.com/paper/1806.02692/) (2018)
- [Learning Traffic Flow Dynamics using Random Fields](https://mlsanity.com/paper/1806.08764/) (2018)
- [Generalized Adaptive Smoothing Using Matrix Completion for Traffic State Estimation](https://mlsanity.com/paper/2206.01461/) (2022)
- [Learning Traffic Speed Dynamics from Visualizations](https://mlsanity.com/paper/2105.01423/) (2021)
- [Generalized adaptive smoothing based neural network architecture for traffic state estimation](https://mlsanity.com/paper/2301.03439/) (2023)
- [Networked Traffic State Estimation Involving Mixed Fixed-mobile Sensor Data Using Hamilton-Jacobi equations](https://mlsanity.com/paper/1606.03332/) (2016)

## Built on by (most cited first, 13 of 13)

- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Spatiotemporal Implicit Neural Representation as a Generalized Traffic Data Learner](https://mlsanity.com/paper/2405.03185/) (2024)
- [Neural operators struggle to learn complex PDEs in pedestrian mobility: Hughes model case study](https://mlsanity.com/paper/2504.18267/) (2025)
- [Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation](https://mlsanity.com/paper/2402.15656/) (2024)
- [Urban traffic analysis and forecasting through shared Koopman eigenmodes](https://mlsanity.com/paper/2409.04728/) (2024)
- [Network-wide Freeway Traffic Estimation Using Sparse Sensor Data: A Dirichlet Graph Auto-Encoder Approach](https://mlsanity.com/paper/2503.15845/) (2025)
- [Physics-informed Partitioned Coupled Neural Operator for Complex Networks](https://mlsanity.com/paper/2410.21025/) (2024)
- [Resolving Extreme Data Scarcity by Explicit Physics Integration: An Application to Groundwater Heat Transport](https://mlsanity.com/paper/2507.06062/) (2025)
- [A novel Fourier neural operator framework for classification of multi-sized images: Application to three dimensional digital porous media](https://mlsanity.com/paper/2402.11568/) (2024)
- [Closed-Loop Neural Operator-Based Observer of Traffic Density](https://mlsanity.com/paper/2504.04873/) (2025)
- [Physics-informed deep operator network for traffic state estimation](https://mlsanity.com/paper/2508.12593/) (2025)
- [DETNO: A Diffusion-Enhanced Transformer Neural Operator for Long-Term Traffic Forecasting](https://mlsanity.com/paper/2508.19389/) (2025)
- [Fractional is Better: Learnable Derivative Orders in Neural Operator Learning](https://mlsanity.com/paper/2610.20363/) (2026)

## Abstract (arXiv)

Deep learning methods are emerging as popular computational tools for solving forward and inverse problems in traffic flow. In this paper, we study a neural operator framework for learning solutions to nonlinear hyperbolic partial differential equations with applications in macroscopic traffic flow models. In this framework, an operator is trained to map heterogeneous and sparse traffic input data to the complete macroscopic traffic state in a supervised learning setting. We chose a physics-informed Fourier neural operator (π-FNO) as the operator, where an additional physics loss based on a discrete conservation law regularizes the problem during training to improve the shock predictions. We also propose to use training data generated from random piecewise constant input data to systematically capture the shock and rarefied solutions. From experiments using the LWR traffic flow model, we found superior accuracy in predicting the density dynamics of a ring-road network and urban signalized road. We also found that the operator can be trained using simple traffic density dynamics, e.g., consisting of 2-3 vehicle queues and 1-2 traffic signal cycles, and it can predict density dynamics for heterogeneous vehicle queue distributions and multiple traffic signal cycles (≥ 2) with an acceptable error. The extrapolation error grew sub-linearly with input complexity for a proper choice of the model architecture and training data. Adding a physics regularizer aided in learning long-term traffic density dynamics, especially for problems with periodic boundary data.