Integrable building blocks for symplectic neural networks: learned-type layers, fiber-preserving layers, and what a single layer can do

Andrew Qing He

Exactly symplectic neural networks are built almost exclusively from shear maps and gain expressivity only through depth. We ask what else a single explicit symplectic layer can be. Observing that every layer in use is the exact flow of a Hamiltonian that is a function of commuting integrals, we obtain three families of layers: a learned-type layer, in which every conjugate pair is rotated, squeezed or sheared by a state-dependent amount and the type is found by training; moving-plane shears, residual symplectic maps outside the known ridge-function class; and fiber-preserving layers, which combine an unconstrained point transformation with a fiber translation at the cost of one linear solve. We characterize residual symplectic layers $x \mapsto x + J\nabla H(x)$: in one degree of freedom they are exactly the shears along a line, while from two degrees of freedom on a strictly larger class exists. Conjugating a learned-type layer by a symplectic network yields an exact one-parameter group with an exactly conserved learned Hamiltonian. In experiments on sixteen systems with one to forty degrees of freedom, against SympNets, HénonNets and learned-plane shear networks, layers built…