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A kernel-based approach to physics-informed nonlinear system identification

Cesare Donati, Martina Mammarella, Giuseppe C. Calafiore, Fabrizio Dabbene, Constantino Lagoa, Carlo Novara

TL;DR

This work advances nonlinear system identification by embedding a physics-based parametric model inside a kernel-based framework, learning a nonparametric correction to account for unmodeled dynamics. The method leverages the representer theorem to obtain a kernel expansion for the unknown term and provides a convex, closed-form solution in the affine-parameter case, while extending naturally to state-space models through nonlinear state smoothing using a combined UKF/URTSS approach. Theoretical results show the kernel correction admits a data-efficient representation and, in favorable cases, enables efficient parameter estimation; numerically, the approach achieves up to $51\%$ reduction in simulation RMSE compared to physics-only models and $31\%$ improvement over state-of-the-art identification techniques. The combination of physics-informed structure with data-driven flexibility yields interpretable models with improved predictive accuracy, validated on academic and cascade-tank benchmarks with robust performance under uncertainty.

Abstract

This paper presents a kernel-based framework for physics-informed nonlinear system identification. The key contribution is a structured methodology that extends kernel-based techniques to seamlessly embed partially known physics-based models, improving parameter estimation and overall model accuracy. The proposed method enhances traditional modeling approaches by embedding a parametric model, which provides physical interpretability, with a kernel-based function, which accounts for unmodeled dynamics. The two models' components are identified from the data simultaneously, thereby minimizing a suitable cost that balances the relative importance of the physical and the black-box parts of the model. Additionally, nonlinear state smoothing is employed to address scenarios involving state-space models with not fully measurable states. Numerical simulations on an experimental benchmark system demonstrate the effectiveness of the proposed approach, achieving up to 51% reduction in simulation root mean square error compared to physics-only models and 31% performance improvement over state-of-the-art identification techniques.

A kernel-based approach to physics-informed nonlinear system identification

TL;DR

This work advances nonlinear system identification by embedding a physics-based parametric model inside a kernel-based framework, learning a nonparametric correction to account for unmodeled dynamics. The method leverages the representer theorem to obtain a kernel expansion for the unknown term and provides a convex, closed-form solution in the affine-parameter case, while extending naturally to state-space models through nonlinear state smoothing using a combined UKF/URTSS approach. Theoretical results show the kernel correction admits a data-efficient representation and, in favorable cases, enables efficient parameter estimation; numerically, the approach achieves up to reduction in simulation RMSE compared to physics-only models and improvement over state-of-the-art identification techniques. The combination of physics-informed structure with data-driven flexibility yields interpretable models with improved predictive accuracy, validated on academic and cascade-tank benchmarks with robust performance under uncertainty.

Abstract

This paper presents a kernel-based framework for physics-informed nonlinear system identification. The key contribution is a structured methodology that extends kernel-based techniques to seamlessly embed partially known physics-based models, improving parameter estimation and overall model accuracy. The proposed method enhances traditional modeling approaches by embedding a parametric model, which provides physical interpretability, with a kernel-based function, which accounts for unmodeled dynamics. The two models' components are identified from the data simultaneously, thereby minimizing a suitable cost that balances the relative importance of the physical and the black-box parts of the model. Additionally, nonlinear state smoothing is employed to address scenarios involving state-space models with not fully measurable states. Numerical simulations on an experimental benchmark system demonstrate the effectiveness of the proposed approach, achieving up to 51% reduction in simulation root mean square error compared to physics-only models and 31% performance improvement over state-of-the-art identification techniques.

Paper Structure

This paper contains 12 sections, 1 theorem, 34 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Kernel-based approximation
  3. Kernel-based model embedding
  4. Affine-in-parameters models
  5. Application to state-space systems
  6. Nonlinear state reconstruction
  7. State-space reformulation
  8. Numerical example
  9. Academic example
  10. Cascade tank system benchmark
  11. Conclusions
  12. Acknowledgments

Key Result

Theorem 1

Consider the same setup of Theorem thm:repthmII. Define $F(x) \doteq [f(x_1), \dots, f(x_T)]^\top \in \mathbb R^{T,n_\theta}$ and $Y_0 \doteq [y_1-f_0(x_1), \dots, y_T-f_0(x_T)]^\top \in \mathbb R^T$. Assume $F(x)$ is full column rank. If the system model in eqn:system.comp is affine in $\theta$, as with where $\mathbf K$ is the kernel matrix associated to $\kappa$ and $\mathcal{D}$, and $\gamma$

Figures (3)

  • Figure 1: Validation RMSE as a function of the kernel bandwidth $\sigma$ and the regularization weight $\gamma$ (log scale), with the optimal hyperparameters $(\sigma^\star, \gamma^\star)$ (magenta dot) selected at the minimum RMSE region.
  • Figure 2: RMSE on validation data as a function of $\gamma$ (log scale).
  • Figure 3: Estimated function and measured data for the test and training datasets.

Theorems & Definitions (7)

  • Definition 1: positive definite kernel scholkopf2001.reprthm
  • Definition 2: reproducing kernel Hilbert space aronszajn1950theorywahba1990spline
  • Remark 1: On the role of $\gamma$
  • Remark 2: On hyperparameter tuning
  • Theorem 1: Closed-form solution of \ref{['eqn:reprthm.ext']}
  • Remark 3: On matrix invertibility and system identifiability
  • Remark 4: estimation without future data