Physics Discovery: SINDy + Conservation Laws + Symbolic Regression

Can the governing physics be recovered from trajectory data alone? Three discovery methods, each answering a different form of that question. SINDy (sequentially-thresholded least squares) reconstructs the explicit equations of motion for Van der Pol, Duffing, and the driven pendulum from a sparse library of candidate terms. A kernel method asks instead what is conserved, extracting invariants through an RBF eigenproblem with no functional form assumed in advance. Symbolic regression by genetic programming rediscovers closed-form laws — kinetic energy, the pendulum period, Ohm’s law — as evolving expression trees. Run on a common benchmark, they separate cleanly into the three epistemic questions of dynamics, invariants, and phenomenological law. Built in Julia (shared custom RK4, no external ODE dependency) as an interactive Pluto notebook.