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

TL;DR

A physics-informed Fourier neural operator ($\pi$-FNO) is chosen as the operator, where an additional physics loss based on a discrete conservation law regularizes the problem during training to improve the shock predictions.

Abstract

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 $(\geq 2)$ 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.

Figures (20)

• Figure 1: Two types of Riemann density solutions possible for LWR-type traffic flow models. The solutions correspond to the condition (a) $u_{l} \leq u_{r} < u_{\rm cr}$ and (a) $u_{r} \leq u_{l} < u_{cr}$, where $u_{\rm cr} = \max_{u} f(u)$ is the critical traffic density.
• Figure 2: Three different types of input conditions considered in this study and their representations: initial condition $\widehat{\mathbf{u}}_{0}$ with domain $\Omega_{0}$, boundary condition $\widehat{\mathbf{u}}_{b}$ with domain $\Omega_{b}$, and interior condition $\widehat{\mathbf{u}}_{p}$ with domain $\Omega_{p}$. $\mathbf{u}$ is the output that defines the solution in the complete domain $\Omega$. Two examples are provided for illustrations.
• Figure 3: The effect of CFL restriction on the computational mesh size in the context of finite volume numerical schemes.
• Figure 4: Architecture of the Fourier neural operator used as the solution operator in the current paper. (a) Complete model architecture. (b) Single Fourier operator layer. These depictions are adapted and modified from li2021fno.
• Figure 5: Sample paths of input conditions drawn from different families of input distributions. The sample paths are parameterized by the distributions of multi-step functions (initial conditions) and multi-wavelet functions (boundary conditions).