Physics-Guided State-Space Model Augmentation Using Weighted Regularized Neural Networks

TL;DR

The paper tackles nonlinear state-space model identification under limited data by augmenting a baseline physics model with a state-space NN. It introduces Weighted PGNN (W-PGNN), which adds a weighted regularization term that penalizes differences in both the state and output functions in regions with low data informativity, using a surrogate input to regularize away from data-rich regions. By weighting the regularization with a neighborhood-based measure, W-PGNN preserves physics in poorly informed regions while allowing data-driven corrections where information is high, and it demonstrates improved generalization over classical PGNN on a 1-D benchmark. The approach advances physics-guided learning by enabling more flexible yet physically consistent model augmentation, with implications for safer and more reliable deployment of data-driven models in control systems.

Abstract

Physics-guided neural networks (PGNN) is an effective tool that combines the benefits of data-driven modeling with the interpretability and generalization of underlying physical information. However, for a classical PGNN, the penalization of the physics-guided part is at the output level, which leads to a conservative result as systems with highly similar state-transition functions, i.e. only slight differences in parameters, can have significantly different time-series outputs. Furthermore, the classical PGNN cost function regularizes the model estimate over the entire state space with a constant trade-off hyperparameter. In this paper, we introduce a novel model augmentation strategy for nonlinear state-space model identification based on PGNN, using a weighted function regularization (W-PGNN). The proposed approach can efficiently augment the prior physics-based state-space models based on measurement data. A new weighted regularization term is added to the cost function to penalize the difference between the state and output function of the baseline physics-based and final identified model. This ensures the estimated model follows the baseline physics model functions in regions where the data has low information content, while placing greater trust in the data when a high informativity is present. The effectiveness of the proposed strategy over the current PGNN method is demonstrated on a benchmark example.

Figures (5)

• Figure 1: Frequency and time responses of two mass spring damper systems with only $10\%$ uncertainties in the system parameters. It can be observed that with a slight shift of resonance frequencies, the outputs are significantly different.
• Figure 2: Model structure of the physics-based SS-NN.
• Figure 3: Illustration of the training, regularization, and test input and output signals used in the simulation.
• Figure 4: Estimation results under the considered approaches in terms of $f_\theta(x)+ax$ (top), and the absolute value of estimation error of $f_\theta(x)$ over $x$ (bottom).
• Figure 5: Test results for the augmented state-space model by the three approaches on the test dataset. The proposed W-PGNN shows the best estimation results both within and outside the training region.