Orthogonal-by-construction augmentation of physics-based input-output models

TL;DR

This work tackles the problem of interpretability and reliable parameter recovery in discrete-time input–output models augmented with a learning component. It introduces an orthogonal-by-construction augmentation that enforces orthogonality between the physics-based baseline and the learning terms using a projection operator $I-\Phi(\Phi^\top\Phi)^{-1}\Phi^\top$, along with a fixed auxiliary parameter $\theta_{aux}=(\Phi^\top\Phi)^{-1}\Phi^\top F^{ANN}_{\theta_a}$. Theoretical results establish exact recovery of the true baseline parameters under identifiability and persistently exciting data, zero covariance between baseline and learning parameters, and estimator consistency as the dataset grows. An identification example with NFIR dynamics demonstrates superior baseline parameter recovery and interpretation, while the learning component accurately captures unmodeled dynamics, highlighting practical benefits for interpretable and extrapolatable physics-informed IO modeling.

Abstract

Model augmentation is a promising approach for integrating first-principles-based models with machine learning components. Augmentation can result in better model accuracy and faster convergence compared to black-box system identification methods, while maintaining interpretability of the models in terms of how the original dynamics are complemented by learning. A widely used augmentation structure in the literature is based on the parallel connection of the physics-based and learning components, for both of which the corresponding parameters are jointly optimized. However, due to overlap in representation of the system dynamics by such an additive structure, estimation often leads to physically unrealistic parameters, compromising model interpretability. To overcome this limitation, this paper introduces a novel orthogonal-by-construction model augmentation structure for input-output models, that guarantees recovery of the physically true parameters under appropriate identifiability conditions.

Figures (4)

• Figure 1: Test errors and the error of the estimated baseline parameters with $\sigma_e=0$ for 10 Monte Carlo runs.
• Figure 2: Estimated baseline parameters with different model structures and training data distributions for $\sigma_e=0$ with 10 Monte Carlo steps. Horizontal dashed lines correspond to the physically true parameter values.
• Figure 3: Panel (a) shows the learning component outputs compared to the true unmodeled terms for 10 Monte Carlo runs. Panels (b) and (c) show the error of the estimated baseline parameters for different training data lengths. The results are averaged over 10 Monte Carlo runs for each $N$ with the bars representing the $\pm$ standard deviation.
• Figure 4: Elements of the asymptotic covariance matrix with (numerically) zero entries shown in black.