PYTHON · NUMPY

Conserve energy by construction

Build a network that cannot violate energy conservation — the Hamiltonian Neural Network — and watch it hold a clean orbit where a black box, trained on the same data, spirals away. This is the structural fix for the drift you saw in Module 2.

01Challenge

In Module 2 you saw a naive integrator invent energy. A black-box network trained to predict dynamics has the same disease, and no loss term fully cures it. The fix is architectural. Instead of predicting the velocities (q̇, ṗ) directly, a Hamiltonian Neural Network learns one scalar — the energy H(q,p) — and DERIVES the motion from it by Hamilton's equations: q̇ = ∂H/∂p, ṗ = −∂H/∂q. A vector field built this way is conservative no matter what the weights are, so it cannot leak energy. The Bench trains an HNN and an ordinary network on the SAME sparse, slightly noisy mass-spring data, then rolls both out from the same point in phase space. Watch: the black box (red) spirals off the true orbit — its energy drifting far — while the HNN (green) stays locked on the gold circle. Same data. The difference is entirely in the wiring.

02Model

A Hamiltonian system's state is (position q, momentum p), and its energy H(q,p) generates the dynamics through Hamilton's equations. The key structural fact: the vector field (∂H/∂p, −∂H/∂q) is divergence-free — it conserves H along every trajectory — for ANY smooth H. So if a network outputs a scalar H and we take its symplectic gradient as the dynamics, conservation is guaranteed by construction, not requested by a penalty. Training matches that derived field to observed velocities; ferromotion computes ∂H/∂q and ∂H/∂p exactly by pushing a two-input jet (value, ∂/∂q, ∂/∂p) through the network on the autodiff tape. The payoff is sharpest with little or noisy data — exactly the physical-AI regime. A black box has to learn the whole 2-D vector field from scattered samples and gets it subtly wrong between them, so its rollout spirals; the HNN only has to learn a scalar landscape, and its field is automatically conservative, so its rollout stays on a closed orbit. This is the deepest form of physics prior: not in the data, not in the loss, but in what the network is even capable of representing.

03Guided practice

Step A (train both): press Train both — a Hamiltonian net and a plain net learn the same 28 noisy samples. Step B (watch the rollout): both start at the same point on the gold true-energy circle. The green HNN point traces the circle round and round; the red black-box point spirals outward (or inward), leaving the orbit. Step C (read the drift): the HNN energy drift stays small (hundredths); the black-box drift grows past the point of doubling the energy. Step D (the lesson): no loss term told the HNN to conserve energy — its architecture made violating conservation impossible. That is what 'physics in the architecture' means.

04Feedback

PASS WHEN PASS when the rollout shows the black box drifting off the true orbit while the HNN conserves energy — a conservation law that came from the architecture, not the data.

If your run fails, check:

Press Train both, then let the rollout run — the red black-box point spirals off the gold orbit while the green HNN point stays on it.
Give it a moment: the energy gap widens as the rollout continues.

05Retrieve & space

Why is the field (∂H/∂p, −∂H/∂q) energy-conserving for ANY choice of H — and how does that guarantee conservation without a loss term?
Why does the HNN's advantage over a black box grow when the data is sparse or noisy?
How is this a deeper form of physics prior than a PINN's loss term?

06Mastery & project

You built a network that conserves energy because of its structure, not because a penalty asked it to — and watched it hold an orbit a black box could not. Physics in the architecture is the strongest of the three insertion points: it constrains what the network can represent at all.

The same idea extends to the full equations of motion: Lagrangian and Deep Lagrangian networks learn the mass matrix and potential while guaranteeing the manipulator-equation structure, and Neural ODEs and Variational Integrator Networks put a structure-preserving integrator in the training loop.

GO FURTHER · THE LIVE PLAYGROUND

ferromotion-learn: the HNN, in pure Rust ↗ — the scalar Hamiltonian + two-input jet you just trained, running on-device.

← Solve an equation with no data Learn the structure, not the map →

PYTHON · NUMPY

Conserve energy by construction

Live ferromotion-wasm: a Hamiltonian net vs a black box on the same sparse mass-spring data. Roll both out and watch which one stays on its energy orbit.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you