Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

IEEE Transactions on Pattern Analysis and Machine Intelligence

Annual Reviews in Control

Traffic state estimation (TSE) bifurcates into two main categories, model-driven and data-driven (e.g., machine learning, ML) approaches, while each suffers from either deficient physics or small data. To mitigate these limitations, recent studies introduced hybrid methods, such as physics-informed deep learning (PIDL), which contains both model-driven and data-driven components. This paper contributes an improved paradigm, called physics-informed deep learning with a fundamental diagram learner (PIDL + FDL), which integrates ML terms into the model-driven component to learn a functional form of a fundamental diagram (FD), i.e., a mapping from traffic density to flow or velocity. The proposed PIDL + FDL has the advantages of performing the TSE learning, model parameter identification, and FD estimation simultaneously. This paper focuses on highway TSE with observed data from loop detectors, using traffic density or velocity as traffic variables. We demonstrate the use of PIDL + FDL to solve popular first-order and second-order traffic flow models and reconstruct the FD relation as well as model parameters that are outside the FD term. We then evaluate the PIDL + FDL-based TSE using the Next Generation SIMulation (NGSIM) dataset. The experimental results show the superiority of the PIDL + FDL in terms of improved estimation accuracy and data efficiency over advanced baseline TSE methods, and additionally, the capacity to properly learn the unknown underlying FD relation.

A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations in the solution set of such systems, in particular only such solutions can be represented by Gaussian process regression. The construction is fully algorithmic via Grobner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.

Linearly Constrained Gaussian Processes with Boundary Conditions

Despite the wide implementation of machine learning (ML) technique in traffic flow modeling recently, those data-driven approaches often fall short of accuracy in the cases with a small or noisy training dataset. To address this issue, this study presents a new modeling framework, named physics regularized machine learning (PRML), to encode classical traffic flow models (referred as physics models) into the ML architecture and to regularize the ML training process. More specifically, leveraging the Gaussian process (GP) as the base model, a stochastic physics regularized Gaussian process (PRGP) model is developed and a Bayesian inference algorithm is used to estimate the mean and kernel of the PRGP. A physics regularizer, based on macroscopic traffic flow models, is also developed to augment the estimation via a shadow GP and an enhanced latent force model is used to encode physical knowledge into the stochastic process. Based on the posterior regularization inference framework, an efficient stochastic optimization algorithm is then developed to maximize the evidence lowerbound of the system likelihood. For model evaluations, this paper conducts empirical studies on a real-world dataset which is collected from a stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform the previous compatible methods, such as calibrated traffic flow models and pure machine learning methods, in estimation precision and is more robust to the noisy training dataset.

Macroscopic traffic flow modeling with physics regularized Gaussian process: A new insight into machine learning applications in transportation

Despite the wide implementation of machine learning (ML) techniques in traffic flow modeling recently, those data-driven approaches often fall short of accuracy in the cases with a small or noisy dataset. To address this issue, this study presents a new modeling framework, named physics regularized machine learning (PRML), to encode classical traffic flow models (referred as physical models) into the ML architecture and to regularize the ML training process. More specifically, a stochastic physics regularized Gaussian process (PRGP) model is developed and a Bayesian inference algorithm is used to estimate the mean and kernel of the PRGP. A physical regularizer based on macroscopic traffic flow models is also developed to augment the estimation via a shadow GP and an enhanced latent force model is used to encode physical knowledge into stochastic processes. Based on the posterior regularization inference framework, an efficient stochastic optimization algorithm is also developed to maximize the evidence lowerbound of the system likelihood. To prove the effectiveness of the proposed model, this paper conducts empirical studies on a real-world dataset which is collected from a stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform the previous compatible methods, such as calibrated pure physical models and pure machine learning methods, in estimation precision and input robustness.

Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications.

Despite the success of classical traffic flow (e.g., second-order macroscopic) models and data-driven (e.g., Machine Learning - ML) approaches in traffic state estimation, those approaches either require great efforts for parameter calibrations or lack theoretical interpretation. To fill this research gap, this study presents a new modeling framework, named physics regularized Gaussian process (PRGP). This novel approach can encode physics models, i.e., classical traffic flow models, into the Gaussian process architecture and so as to regularize the ML training process. Particularly, this study aims to discuss how to develop a PRGP model when the original physics model is with discrete formulations. Then based on the posterior regularization inference framework, an efficient stochastic optimization algorithm is developed to maximize the evidence lowerbound of the system likelihood. To prove the effectiveness of the proposed model, this paper conducts empirical studies on a real-world dataset that is collected from a stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform the previous compatible methods, such as calibrated physics models and pure machine learning methods, in estimation precision and input robustness.

Xianfeng Terry Yang

Papers

Macroscopic traffic flow modeling with physics regularized Gaussian process: A new insight into machine learning applications in transportation

Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications.

Highway Traffic State Estimation Using Physics Regularized Gaussian Process: Discretized Formulation.

Examining driver injury severity in left-turn crashes using hierarchical ordered probit models.