Theory Foundation of Physics-Enhanced Residual Learning

Intensive studies have been conducted in recent years to integrate neural networks with physics models to balance model accuracy and interpretability. One recently proposed approach, named Physics-Enhanced Residual Learning (PERL), is to use learning to estimate the residual between the physics model prediction and the ground truth. Numeral examples suggested that integrating such residual with physics models in PERL has three advantages: (1) a reduction in the number of required neural network parameters; (2) faster convergence rates; and (3) fewer training samples needed for the same computational precision. However, these numerical results lack theoretical justification and cannot be adequately explained. This paper aims to explain these advantages of PERL from a theoretical perspective. We investigate a general class of problems with Lipschitz continuity properties. By examining the relationships between the bounds to the loss function and residual learning structure, this study rigorously proves a set of theorems explaining the three advantages of PERL. Several numerical examples in the context of automated vehicle trajectory prediction are conducted to illustrate the proposed theorems. The results confirm that, even with significantly fewer training samples, PERL consistently achieves higher accuracy than a pure neural network. These results demonstrate the practical value of PERL in real world autonomous driving applications where corner case data are costly or hard to obtain. PERL therefore improves predictive performance while reducing the amount of data required.