The field of dynamical systems and partial differential equations (PDEs) is witnessing significant advancements, driven by the development of innovative methods and techniques. One key area of focus is the optimization of trajectories for high-dimensional robotic systems, where researchers are exploring new approaches to improve computational efficiency and accommodate arbitrary cost functions. Another area of interest is the development of scalable and interpretable neural surrogates for solving high-dimensional PDEs, which is crucial for various scientific and engineering applications. Recent works have also highlighted the potential of integrating neural operators with advanced numerical methods to perform system-level analysis and stability assessments. Noteworthy papers include:

• A study that generalizes the Affine Geometric Heat Flow formulation to accommodate arbitrary cost functions, enabling the generation of dynamically feasible trajectories for complex systems.
• A paper that introduces Anant-Net, a neural surrogate that efficiently solves high-dimensional PDEs, achieving high accuracy and robustness across various benchmark tests.
• A work that proposes a framework for integrating local neural operators with advanced iterative numerical methods to perform efficient system-level stability and bifurcation analysis of large-scale dynamical systems.