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.
Abstract
Ordinary differential equations (ODEs) are widely used to model dynamical systems in physics, biology, neuroscience, and physiology, but in many applications some equations of the dynamics are unknown and only a subset of the state variables are measured. We propose 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. The proposed method consists of two stages where we alternate between state and parameter estimation and iterate until a predetermined criterion is met. Specifically, in the first step, we treat the model parameters as being known and we infer the latent states from the available measurements using a Rauch--Tung--Striebel (RTS) smoother. In the second stage, we treat the smoothed trajectories as being known and use them to estimate the neural networks'parameters through backpropagation. We evaluate the method on benchmark systems spanning linear, nonlinear, and stiff dynamics under partial state observation. Across these settings, the proposed method learns missing ODE components from incomplete measurements while exploiting and retaining interpretable mechanistic structure and improving latent-state reconstruction and long-horizon prediction.
Spectral submanifolds can be used to reduce recurrent neural networks to low-dimensional models, revealing their core dynamics, and authors uncover robust structures underlying decision-making and working-memory tasks, providing predictions about the underlying behavior of neural computations.
A. Marraffa, R. Krause, V. Mante et al.· Nature Communications· 0 citations
For numerical investigation of dynamical systems, the formulation of the corresponding ordinary differential equations (ODE) based on physical principles is usually the first and most crucial step. However, if the underlying physics is not fully understood or the required expert knowledge for modeling is missing, setting up these differential equations fail. Sometimes, running either real‐world experiments or black‐box simulations with commercial of‐the‐shelf software are the only ways of system exploration, which can be time‐consuming and/or expensive. In such cases, based on the gathered data, a surrogate model for the ODE may be set up and trained with the goal of later substituting the missing differential equation for cheaper numerical investigations.
In this paper, an approach is presented which proposes the embedding of a neural network based architecture as a substitute for the state function of an ODE into established Runge–Kutta based integration method. For training the surrogate, the numerical integration method solves an initial value problem to map initial conditions onto target system states for comparison with training data. The corresponding loss is then backpropagated through the model graph spanned by the numerical integration scheme to update the adjustable weights of the neural network for minimizing the loss of the mapping. The optimized surrogate state function may finally be treated as a substitute for the differential equation under investigation.
Timo Bielitz, Dieter Bestle· Proceedings in Applied Mathe...· 0 citations
A Bi-stage Gaussian Process (GP) framework for non-autonomous ODEs, capable of estimating system states and their derivatives directly from noisy measurements, providing a reliable alternative to classical ODE modeling in noisy and complex systems.
R. Fezai, Byanne Malluhi, Nour Basha et al.· IEEE Access· 0 citations
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.
Laurin Ludmann, Jaeyoun Choi, J. Neubeck et al.· Vehicles· 0 citations
The Physics-Informed Stochastic Configuration Machine is proposed, a novel backpropagation-free framework for both forward and inverse problems in differential equations that achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs.
Yueze Song, Zhong-Zhe Chen, Li-Hui Cen et al.· 0 citations
Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Rishav Jha, Kameshwar Sahani, S. K. Sahani et al.· African Multidisciplinary Jo...· 0 citations
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