📄 AI Poincaré: Machine Learning Conservation Laws from Trajectories

Summary

The paper asks whether conservation laws can be found purely from observed trajectories, without knowing the equations of motion. The authors build on a geometric idea: all states a trajectory visits lie on a manifold in phase space, and each conserved quantity reduces that manifold's dimension by one. Counting conserved quantities therefore becomes a dimensionality-estimation problem.

AI Poincaré works in three stages:

• Preprocessing: the data are prewhitened, and any directions with zero variance are removed as linear invariants.
• Walk-pull Monte Carlo: points are perturbed with Gaussian noise of scale L, and a neural network, following neural empirical Bayes, learns to project them back onto the manifold.
• Dimension estimation: PCA of the pulled-back samples, tracked across noise scales, produces an "explained ratio diagram." Components that nearly vanish signal conservation laws.

On five Hamiltonian benchmarks, including the gravitational three-body problem, the method recovered the correct number of conserved quantities in every case. It also exposed periodic orbits, phase transitions and the timescales on which approximate conservation fails. This matters as a step toward fully automated, data-driven discovery in physics.

Key points

• Test systems: the 1D harmonic oscillator, 2D Kepler problem, double pendulum, 2D magnetic mirror and 2D three-body problem. Trajectories were generated with a 4th-order Runge-Kutta integrator.
• Pull network and sampling: a feedforward net with two hidden layers of 256 units, trained with ADAM (learning rate 10⁻³) for 5,000 steps per noise scale. Each run used 10⁴ walk-pull jumps.
• Detection criterion: a conservation law is counted when a principal component explains less than 0.1/N of the variance. Valleys in the explained ratio diagram consistently appear near L ≈ 0.1.
• Three-body problem: the four linear invariants (center-of-mass position and momentum) were already caught during preprocessing.
• Symbolic formulas: combining the method with AI Feynman yielded explicit expressions for 10 of the 13 conserved quantities. The three misses were the Runge-Lenz vector, the large-angle double-pendulum energy and the three-body energy.