First Integral

First integrals, representing quantities conserved during a system's evolution, are a central concept in dynamical systems and related fields. Current research focuses on discovering and utilizing first integrals from data, employing methods like sparse regression (SINDy), neural networks (including neural differential equations), and integral projection-based autoencoders, often integrated with other techniques such as model predictive control or symbolic regression. These advancements enable more robust parameter estimation, improved model interpretability, and enhanced predictive capabilities across diverse applications, ranging from physics simulations and chemical process optimization to robotics and image analysis. The ability to efficiently learn and utilize first integrals promises significant improvements in modeling complex systems and solving challenging scientific problems.