Symplectic Geometry on Chen Kai Blog

Symplectic Geometry and Structure-Preserving Neural Networks

Mon, 28 Jul 2025 09:00:00 +0000

Train a vanilla feedforward network to predict a one-dimensional harmonic oscillator. Validate it on the next ten time steps — the error is fine. Now roll it out for a thousand steps. The orbit no longer closes, the energy creeps upward, and what should be periodic motion turns into a slow spiral. The network learned to fit data points but never learned the physics. Structure-preserving networks fix this by incorporating geometric invariants — energy conservation, the symplectic 2-form, and the Euler-Lagrange equations — directly into the architecture, ensuring the learned model cannot violate them no matter how long you integrate.

PDE and ML (5): Symplectic Geometry and Structure-Preserving Networks

Sun, 30 Jun 2024 09:00:00 +0000

A pendulum keeps swinging for a very long time without slowly winding down — energy is conserved. The Earth orbits the Sun for billions of years without flying off — angular momentum is conserved. Behind every “this quantity stays constant” lurks a piece of geometry called symplectic structure.

Train a vanilla Neural ODE on pendulum data: after a few hundred steps the energy drifts. The network can fit the short-term trajectory just fine; what it can’t fit is the long-time conservation law. Structure-preserving networks (HNN, LNN, SympNet) take a different approach: bake the conservation law into the architecture so the network cannot violate it.