Advances in Nonlinear Dynamics and Stochastic Systems

The field of nonlinear dynamics and stochastic systems is witnessing significant developments, with a focus on improving the accuracy and reliability of system identification, modeling, and analysis. Researchers are exploring new approaches to address the challenges of noise sensitivity, rational functions, and complex nonlinearities in mechanical systems. The integration of machine learning and control theory is also gaining traction, enabling real-time learning and inference of complex, time-varying nonlinear models. Furthermore, advancements in numerical methods and algorithms are being made to efficiently compute invariant measures, estimate intrinsic noise, and solve highly oscillatory ordinary differential equations. Noteworthy papers in this area include: The paper on differentiable sparse identification of Lagrangian dynamics, which presents a novel framework for accurate representation of complex nonlinearities and robust equation discovery. The paper on on-line learning of dynamic systems, which introduces a unified framework combining sparse regression and Kalman filtering for real-time inference of nonlinear models. The paper on learning stochasticity, which proposes a nonparametric framework for intrinsic noise estimation in dynamical systems.

Sources

Differentiable Sparse Identification of Lagrangian Dynamics

On-line learning of dynamic systems: sparse regression meets Kalman filtering

Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

Data-driven computation for periodic stochastic differential equations

Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity

Learning stochasticity: a nonparametric framework for intrinsic noise estimation

Error Analysis on a Novel Class of Exponential Integrators with Local Linear Extension Techniques for Highly Oscillatory ODEs

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