The paper proposes a new interpretable machine learning approach for discovering unknown nonlinear ordinary differential equations from a single state trajectory. It differs from existing methods by deriving its formulation from Functional Analysis and Operator Theory and by defining a cost function as an integral distance between functions rather than a discrete error sum. An incremental learning algorithm enables online updates, allowing simultaneous identification of both system dynamics and external time‑varying forces, with numerical examples illustrating its benefits.
arXiv:2606. 08956v1 Announce Type: new Abstract: Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature.
arXiv:2607. 05280v1 Announce Type: new Abstract: Many real-world systems evolve continuously, yet most machine learning models interpret time series as discrete sequences.
arXiv:2412. 12036v2 Announce Type: replace Abstract: System identification, the process of deriving mathematical models of dynamical systems from observed input-output data, has undergone a paradigm shift with the advent of learning-based methods.
arXiv:2410.23667v2 Announce Type: replace Abstract: Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known con...
arXiv:2609.37083v1 Announce Type: cross Abstract: We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing...
arXiv:2606. 09638v1 Announce Type: new Abstract: Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena.
arXiv:2607. 15180v1 Announce Type: new 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.
Identifying the governing equations of complex dynamical systems remains a fundamental challenge across science and engineering. While early approaches relied on empirical data and heuristics, modern data-driven methods offer greater flexibility and fewer assumptions.
arXiv:2608.22112v1 Announce Type: cross Abstract: We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approac...
The paper introduces Dynafit, a kernel-based method for classifying trajectories produced by distinct nonlinear dynamical systems. It learns a distance metric by approximating the Koopman operator, enabling classification in a feature space without explicit dimensionality. The authors demonstrate Dynafit on logistic map chaos detection, handwritten dynamical pattern recognition, and visual dynamic texture classification.
arXiv:2511. 08860v2 Announce Type: replace-cross Abstract: The deep learning revolution has spurred a rise in advances of using AI in sciences.
This paper introduces a data‑driven neural power‑iteration algorithm for approximating the dominant eigenfunctions (modes) of the Koopman operator in nonlinear dynamical systems. By avoiding explicit construction of the operator’s projection, the method sidesteps the curse of dimensionality that plagues expressive neural templates. The authors provide theoretical convergence guarantees tied to sample size and network width, and demonstrate through numerical experiments that the approach yields accurate, smooth mode approximations without the drawbacks of traditional techniques such as extended dynamic mode decomposition.