How Machines Learn the Structure of Reality

Physical laws are not mere arbitrary regression fits—they reflect deep symmetry groups and conservation principles (Noether's Theorem). When neural architectures or symbolic engines inspect physical trajectories, they identify invariants: quantities that remain precisely constant along the system's vector field.

This workbench implements SINDy (Sparse Identification of Nonlinear Dynamics):

• Phase Reconstruction: We record coordinate-momentum pairs $\mathbf{x}(t) = [q(t), p(t)]$ and compute finite-difference velocity vectors $\dot{\mathbf{x}}(t)$.
• Nonlinear Library Matrix $\Theta(\mathbf{X})$: We build candidate basis functions: $[1, q, p, q^2, qp, p^2, q^3, \sin(q), \cos(q)]$.
• Sparse Ridge Thresholding: We solve $\dot{\mathbf{x}} = \Theta(\mathbf{x})\mathbf{\Xi}$ via sequential thresholded least squares, zeroing coefficients below $\lambda$. Parsimony drives the model to choose the exact natural law.
• Symplectic Invariant Recovery: By computing the curl and verifying $\frac{\partial H}{\partial p} = \dot{q}$ and $-\frac{\partial H}{\partial q} = \dot{p}$, the algorithm reconstructs the system's Hamiltonian without human supervision.

Understanding Conserved Quantities

Try selecting different dynamical systems above:

• Nonlinear Pendulum: Transition from linear harmonic oscillation at low energy to large-amplitude nonlinear periods, where $\sin(q) \neq q$.
• Kepler 2-Body: Inverse-square gravitational orbits conserving both total energy $E$ and angular momentum $L = r^2 \dot{\theta}$.
• Duffing Oscillator: A double-well potential exhibiting bistability and chaotic separatrix crossing under perturbation.
• Lotka-Volterra: Ecological predator-prey dynamics preserving a non-polynomial logarithmic conserved manifold $V(x,y)$.