The experimental results in this application confirm that hybrid models yield superior performance over strict physical approaches and can implicitly approximate submodel dynamics within a unified, yet modular, architecture, opening avenues for applications in domains where partial physics-based knowledge is available but insufficient on its own.
Abstract
This paper introduces Hybrid Physics-Augmented Neural Network (HyPA-Net), a hybrid modeling framework that integrates physics-based linear time-invariant models with artificial neural networks (ANNs) to address dynamic system modeling when only partial physical knowledge is available. The approach leverages the interpretability and robustness of established physical models while using ANNs—such as long short-term memory architectures—to capture unknown or nonlinear system behaviors. The methodology normalizes state and input variables for compatibility with ANN training and expands traditional recursive state-space equations for efficient backpropagation over sequences. Vehicle dynamics, specifically using a rear-wheel steering test case, validate the proposed framework. Various HyPA-Net configurations are benchmarked against pure physics-based and pure data-driven models, demonstrating improved prediction accuracy and model flexibility. The experimental results in this application confirm that hybrid models yield superior performance over strict physical approaches and can implicitly approximate submodel dynamics within a unified, yet modular, architecture, opening avenues for applications in domains where partial physics-based knowledge is available but insufficient on its own.
Physics-Informed Neural Networks (PINNs) aim to incorporate the phenomenology of the process into their training through an additional mathematical model based on conservation principles (physical laws). This work presents a method for training feedforward neural networks with physics-informed constraints related to static gain signals between the output and inputs in real cases in which a phenomenological model is not available. The proposed approach was applied to feedforward networks considering gradient-based and gradient-free learning methods (Extreme Learning Machine, ELM). This work shows that, even in training approaches involving a weight initialization strategy coupled with a constructive algorithm for defining the number of hidden units, the neural model, identified without any physical information, does not ensure that static gain signals between the output and inputs are consistent with the physical reality of the phenomenon. The case studies comprised 3 real datasets widely used as benchmarks for regression problems. The results show that in the absence of any equations capable of describing the physics of the analyzed problem, it is feasible and desirable to incorporate hard constraints into the training. This can ensure the correct direction of effect of each input on each output, and therefore the qualitative consistency of the models, besides their quantitative performances.
Conventional integer-order models do not adequately represent intrinsic memory effects in the dynamics of modern power systems, which operate under increasingly nonlinear and uncertain conditions. This work proposes a Fractional-Order Physics-informed Neural Network (FOPINN) framework for power system stability analysis and forward dynamic modelling. The method allows for an accurate representation of memory-dependent dynamics by directly integrating fractional-order swing equations into the learning process. The framework is extended to a reduced Multi-Machine Infinite Bus (MMIB) system to examine inter-machine coupling and synchronisation dynamics, following with a validation on a Single Machine Infinite Bus (SMIB) system. A comparison with integer-order PINNs, Caputo-L1 fractional solvers, and classical RK4 shows that FOPINN achieves significantly improved prediction accuracy while maintaining physical consistency. The trained FOPINN further provides approximately $193\times $ faster inference than repeated Caputo–L1 numerical computation while demonstrating the expected influence of fractional order, damping, inertia, and network coupling on transient stability. The results show that transient behaviour is strongly influenced by fractional-order dynamics. Stronger memory effects alter the response as the fractional order decreases, causing long-lasting transient deviations and slower convergence before the system reaches steady state. It is demonstrated that memory effects propagate via machine coupling in the MMIB system, altering synchronisation characteristics and the stability margin as the disturbance level increases. Fractional-order analysis reveals oscillatory or unstable responses in high-loading regimes, whereas classical integer-order models predict stable behaviour. These results establish FOPINN as an effective, reusable and physics-consistent framework for capturing memory-driven dynamics in contemporary power grids and emphasise the significance of integrating fractional-order modeling for realistic power system analysis.
V. S. Malavika, E. Gopalakrishnan, K. Chandan et al.· IEEE Access· 0 citations
A Decoupled Hybrid Residual Model for online adaptive prediction of vehicle dynamics is proposed, demonstrating that the proposed architecture effectively improves prediction accuracy, robustness, and implementation feasibility under varying driving conditions.
Guodong Zhu, Jialing Yao, Yiwen Bai et al.· Proceedings of the Instituti...· 0 citations
A hybrid neural--physics framework in which the known components of the ODE are kept explicit and the missing components are represented by a neural network to improve latent-state reconstruction and long-horizon prediction.
Ahmet Demirkaya, Georgios Stratis, Tales Imbiriba et al.· arXiv.org· 0 citations
Hybrid computational models can improve prediction when governing equations are incomplete, but their advantages are often evaluated on isolated systems and without simultaneous assessment of accuracy, stability, interpretability, and uncertainty. This study develops a modular hybrid mathematical model that combines a partially specified ordinary differential equation, a regularized neural residual, joint parameter calibration, physical constraints, and ensemble-based uncertainty quantification. The framework was evaluated through in silico experiments on five benchmark systems representing periodic, chaotic, stiff, ecological, and engineering dynamics: Van der Pol, Lorenz-63, Robertson kinetics, Lotka–Volterra, and a continuous stirred-tank reactor. Six hundred trajectories were generated using space-filling sampling, partitioned at the trajectory level, and tested under interpolation, extrapolation, measurement noise, data scarcity, and partial observability. The proposed model was compared with an incomplete mechanistic model, a neural ordinary differential equation, and sequential residual correction. In illustrative synthetic results, the hybrid model achieved a mean normalized root-mean-square error of 0.0678, reducing error by 37.3% relative to the strongest baseline. It also increased the mean time to divergence to 83.6% of the forecast horizon, reduced median mechanistic parameter error to 4.8%, limited physical-constraint violations to 0.6%, and attained 0.947 coverage for nominal 95% prediction intervals. Friedman and Holm-adjusted Wilcoxon tests indicated significant paired improvements with large effect sizes. Ablation analyses showed that residual regularization, physical constraints, and joint calibration each contributed materially. These findings illustrate how restricted data-driven correction can enhance heterogeneous dynamical-system prediction while preserving mechanistic meaning, although empirical execution and external validation are required before scientific claims are made
George Em Karniadakis, E. Torfs, L. Marchetti· Global Synthesis in Educatio...· 0 citations
Neural network techniques have been widely exploited to model structural dynamics. Among them, the continuous-time state-space neural network (CSNN) possesses great potential because of its advantage that a trained CSNN model can operate at different sampling rates without retraining. However, it has relatively low training efficiency due to the integration operations involved. To address this limitation, this study proposes a Wiener-type neural network (WNN) based on CSNN by reducing the nonlinear state derivative calculator present in CSNN to a linear equation. Based on this modification, an explicit state expression for WNN that discards high-order differentiable items for back propagation is derived, which not only facilitates rapid computation of the state variable but also greatly enhances training efficiency. The effectiveness of WNN is assessed through a numerical example of a cubic-stiffness structure, an on-site measurement example of a 6-story hotel building, and an experimental example of a magneto-rheological fluid damper. WNN models are compared with different models for these examples, and the results indicate that WNN models achieve high and consistent prediction accuracy, with Pearson correlation coefficients exceeding 0.85 and normalized root mean square errors below 0.06 across all cases. Meanwhile, the WNN model exhibits up to an 83% reduction in training time relative to the CSNN model. The developed WNN effectively balances prediction accuracy and training efficiency, making it a promising approach for dynamic modeling of large and complex systems.
Hongwei Li, Yao Hu, Panpan Gai et al.· International Journal of Str...· 0 citations
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