Vehicle single track modeling using physics guided neural differential equations

TL;DR

The paper addresses the challenge of accurate, real-time vehicle dynamics by integrating physics with data-driven learning through physics guided neural differential equations. It compares a white-box ODE, a black-box neural ODE, and a hybrid universal differential equation (UDE) approach for a single-track drift problem, showing that the UDE achieves the best validation performance while using a smaller neural network. The key contributions are a systematic comparison across modeling paradigms, demonstration that the hybrid UDE can reduce the neural network size by an order of magnitude while improving generalization, and practical guidance on training strategies such as multiple shooting. The findings indicate that physics-guided hybrid models can deliver accurate, data-efficient, and deployable dynamics models for automotive applications, with potential impact on real-time control and state estimation in autonomous systems.

Abstract

In this paper, we follow the physics guided modeling approach and integrate a neural differential equation network into the physical structure of a vehicle single track model. By relying on the kinematic relations of the single track ordinary differential equations (ODE), a small neural network and few training samples are sufficient to substantially improve the model accuracy compared with a pure physics based vehicle single track model. To be more precise, the sum of squared error is reduced by 68% in the considered scenario. In addition, it is demonstrated that the prediction capabilities of the physics guided neural ODE model are superior compared with a pure black box neural differential equation approach.

Figures (7)

• Figure 1: Top view of vehicle single track and single track drift model. The dot in center of the figure represents the center of gravity.
• Figure 2: Exogenous input of acceleration ($a_x(t)$) and steer angle velocity ($v_\delta (t)$) of data sample three in top and bottom panel respectively. The x-axis denotes simulation time in seconds.
• Figure 3: Estimated state trajectories of ODE model (red line) and reference model (black line) over the training batch. Note the large deviation between ODE mode and reference data in the velocity $v$ (fifth panel).
• Figure 4: Validation error of all neural ODE (nODE) and UDE model over the number of neural network weights. The subscripts in the model names denote the hidden layer size. For instance, UDE10 means the UDE model with ten neurons in the hidden layer. Dots connected with vertical lines denote repeated trainings on different initialization of the network weights. Please note the logarithmic scale of the validation error.
• Figure 5: Estimated state trajectories of neural ODE10 model with 10 neurons in hidden layer (red line) and reference model (black line). The vertical line at $t=70\,\mathrm{s}$ marks the split between training and validation data.