The field of complexity theory and dynamical systems is witnessing significant developments, with a growing emphasis on innovative methods for modeling and analyzing complex systems. Researchers are exploring new approaches to parameterize and learn the dynamics of integrable functions, discrete dynamical systems, and fractional-order nonlinear systems. Notably, hybrid machine learning schemes and data-integrated frameworks are being proposed to improve the accuracy and efficiency of reduced-order models. Additionally, there is a focus on preserving the Hamiltonian structure in reduced models to maintain long-term stability. These advancements have the potential to impact various applications, including plasma dynamics and the treatment of Parkinson's disease.

Noteworthy papers include:

• A Hybrid Neural Network -- Polynomial Series Scheme for Learning Invariant Manifolds of Discrete Dynamical Systems, which proposes a hybrid scheme that combines polynomial series with shallow neural networks to learn invariant manifolds.
• Reduced Particle in Cell method for the Vlasov-Poisson system using auto-encoder and Hamiltonian neural, which introduces a nonlinear, non-intrusive, data-driven model order reduction method that preserves the Hamiltonian structure.
• A Data-Integrated Framework for Learning Fractional-Order Nonlinear Dynamical Systems, which presents a framework for learning the dynamics of fractional-order nonlinear systems using input-output experiments and orthonormal basis functions.