Physics-Informed Training Strategies for Neural Estimators in ODE-governed Dynamical Systems: an Application to Vehicle Sideslip Estimation

For decades, state estimation has been a challenging problem in many fields, including vehicle dynamics. Classical approaches such as model-based estimators or data-driven neural networks come with important limitations. Hybrid methodologies that combine model-based and machine learning techniques have recently been proposed, with Physics-Informed Neural Networks (PINNs) potentially offering substantial benefits. To this end, this paper proposes strategies for embedding physical knowledge into the training of neural networks for state estimation in systems governed by ordinary differential equations (ODEs), with focus on vehicle sideslip angle estimation. The methodologies herein investigated are: i) physical regularization, consisting of augmenting the data-driven loss with a physics-based term, with the introduction of a novel adaptive regularization scaling factor, denoted as fidelity; ii) a novel methodology, denoted as physical learning, that combines supervised data-driven learning with unsupervised physical learning, leveraging unlabelled datasets (that require no ground-truth measurements). The two methodologies are proposed and validated on both feed-forward and long short-term memory (LSTM) architectures. Furthermore, the paper addresses technical aspects of PINNs often overlooked yet significantly affecting the network performance, including the computation of output derivatives, the formulation of the physical loss, and the selection of input and batch features. A key contribution of this work lies in the generality of these methodologies, which extend beyond vehicle sideslip estimation to a broad class of ODE-governed state estimation problems. Comparative analyses on experimental vehicle data demonstrate that both methodologies improve generalization compared to their standard data-driven counterparts, with physical learning showing particularly strong potential - with a reduction up to 38% in terms of mean squared error (MSE) for feedforward architectures and 44% for LSTM-based architectures in a dry-asphalt testing scenario, and up to 82% in a icy-asphalt scenario without labelled data.