Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints

Reasoning about the physical world requires models that are endowed with the right inductive biases to learn the underlying dynamics. Recent works improve generalization for predicting trajectories by learning the Hamiltonian or Lagrangian of a system rather than the differential equations directly. While these methods encode the constraints of the systems using generalized coordinates, we show that embedding the system into Cartesian coordinates and enforcing the constraints explicitly with Lagrange multipliers dramatically simplifies the learning problem. We introduce a series of challenging chaotic and extended-body systems, including systems with N -pendulums, spring coupling, magnetic fields, rigid rotors, and gyroscopes, to push the limits of current approaches. Our experiments show that Cartesian coordinates with explicit constraints lead to a 100x improvement in accuracy and data efficiency. Gyroscope system 260x 100x Gyroscope Hamiltonian Cartesian coordinates (easy to learn) Angular coordinates (hard to learn) Data-efficiency & accuracy H(X, P ) = 1 2 Tr(P M -1 P ) + gmX03 M -1 ii = (1 + 1/λi)/m for i = 1, 2, 3 M -1 00 = M -1 ij = 1/m for i = j H(q, p) = 1 2 p T M (q) -1 p + mg cos q2 M11 = sin 2 q2(I1 sin 2 q3 + I2 cos 2 q3) + cos 2 q2I3 M12 = M21 = (I1 -I2) sin q2 sin q3 cos q3 M33 = I3 M13 = M31 = I3 cos q2 M22 = I1 cos 2 q3 + I2 sin 2 q3 * Equal contribution. 34th Conference on Neural Information Processing Systems (NeurIPS 2020),