eess.SYOct 12, 2026

Scalable Identification of Physics-Based and Neural Dynamics from Partial Measurements

Authors: Yibo Zhou, Johannes Köhler

Organizations: Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden · Department of Mechanical Engineering, Imperial College London, London, United Kingdom

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Abstract

We present a scalable method for identifying nonlinear dynamics from partial measurements, supporting physics-based, neural-network, and gray-box models. We use the Prediction Error Method (PEM) coupled with state estimation performed by a Moving Horizon Estimation (MHE) scheme, which can account directly for general nonlinear dynamics. This results in a bi-level optimization problem that scales poorly with increasing model parameters. To improve computational efficiency, we derive a reformulation that avoids explicit parameter-wide sensitivity matrices and exploits the temporal structure of the underlying optimization problem. Numerical comparisons show up to a 215×215\times speedup in hypergradient evaluation. Racing-vehicle simulations demonstrate accurate prediction and physical-parameter recovery in tire-force identification. Real-world data show that the gray-box model improves prediction accuracy with a neural residual while retaining physical interpretability.

Figures & tables

Figure 1: Vehicle geometry and modeling conventions. Schematic adapted from Chrosniak et al. (2024) .
Figure 2: Median hypergradient computation time over 100 MHE windows versus NN parameter count nNNn_{\mathrm{N}N} .
MethodOne-Step Prediction ErrorAverage Relative Physical-Parameter ErrorTraining Time
 TrainValidation 
Ground True Parameters6.27×10−146.27\times 10^{-14}9.71×10−149.71\times 10^{-14}00–
SUBNET2.85×10−22.85\times 10^{-2}1.66×10−11.66\times 10^{-1}–2 min
Joint-MHE1.37×10−21.37\times 10^{-2}4.42×10−34.42\times 10^{-3}69.05%69.05\%7 min
EKF →\rightarrow PEM1.33×10−41.33\times 10^{-4}6.49×10−56.49\times 10^{-5}22.99%22.99\%1 min
EKF →\rightarrow PEM (adaptive)1.60×10−41.60\times 10^{-4}9.52×10−59.52\times 10^{-5}161.93%161.93\%26 min
Table 1: Identification results on vehicle data simulated with BayesRace, comparing one-step prediction error, physical-parameter recovery, and training time. Best and worst error values are highlighted in green and red, respectively.
MethodOne-Step Prediction ErrorTraining Time
 TrainValidation
SUBNET3.16×10−43.16\times 10^{-4}3.25×10−33.25\times 10^{-3}3 min
Joint-MHE4.70×10−44.70\times 10^{-4}4.47×10−44.47\times 10^{-4}4 min
EKF →\rightarrow PEM4.37×10−44.37\times 10^{-4}4.20×10−44.20\times 10^{-4}1 min
EKF →\rightarrow PEM (adaptive)2.20×10−42.20\times 10^{-4}2.62×10−42.62\times 10^{-4}4 min
PEM-MHE (physics-only)4.24×10−44.24\times 10^{-4}4.10×10−44.10\times 10^{-4}2 h 50 min
Table 2: Identification results on the IAC real-world vehicle dataset, comparing one-step prediction error and training time. Best and worst error values are highlighted in green and red, respectively.
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