An interpretable identification framework based on universal neural ordinary differential equations (UNODEs), symbolic regression, and parameter refinement is developed that is competitive on autonomous polynomial systems and more effective in recovering compact symbolic structures for non-polynomial and explicitly time-varying dynamics.
Abstract
Interpretable data-driven techniques are of significant research value in dynamical system identification. This paper develops an interpretable identification framework based on universal neural ordinary differential equations (UNODEs), symbolic regression, and parameter refinement. In the proposed framework, UNODEs are used as a front-end model to learn state- and time-dependent vector fields, while symbolic regression is employed to transform the learned black-box vector field into an explicit governing equation. After the symbolic structure is identified, model parameters are further refined at the trajectory level to improve the numerical consistency of the mechanistic model. Theoretical justification for the uniqueness and identifiability of the recovered model is provided via the regularized Neural ODE framework. Comparative experiments against SINDy, PNODE, and ODENet demonstrate that the proposed method is competitive on autonomous polynomial systems and more effective in recovering compact symbolic structures for non-polynomial and explicitly time-varying dynamics.
The data-driven discovery of governing equations for dynamical systems has emerged as a transformative paradigm, enabling the extraction of interpretable and generalizable models from observational data. While modern techniques have advanced this field, traditional subset regression remains a foundational yet underutilized tool due to its reliance on uncorrelated residuals, a requirement often violated by time-series data. In this work, we revisit subset regression to identify dynamical systems governed by ordinary differential equations (ODEs), partial differential equations (PDEs), and differential algebraic equations (DAEs). We propose subset regression with known number of active features (sub-KNAFE), a user-determined sparsity mechanism that flexibly adapts to various complex nonlinear systems, while retaining the computational efficiency and inherent interpretability of traditional subset regression. We integrate sub-KNAFE with the SINDy framework, overcoming the limitation of subset regression in dynamical system identification. Numerical tests across a range of signal-to-noise ratios and dataset sizes demonstrate sub-KNAFE's superior noise robustness and data efficiency. Practical utility for sub-KNAFE is validated on two real-world datasets: the classic Lynx-Hare ecological population data and the ISO New England power system dataset, demonstrating its strong potential for practical deployment in scientific discovery and engineering applications.
Weizhen Li, Qiang Fu, Yifan Hong et al.· Scientific Reports· 0 citations
Multi-step training of sparse, interpretable models of dynamical systems directly from time-series data yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents.
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
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
This review surveys system identification for complex dynamical systems through a unified framework that connects model structure, operating mode, estimation target, and computational setting and identifies key open challenges, including robust identification under biased or limited data, uncertainty-aware modeling, interpretable learning, reproducible benchmarking, and tighter integration between identification, experiment design, and control synthesis.
Xiaoyu Zhang, Zi-Qiang Li, Ruizhe Shi et al.· Complex Engineering Systems· 0 citations
This thesis advances the training and scalability of NCDEs through three complementary contributions, building on neural rough differential equations, which reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.
Benjamin Walker· 1 citation
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