PYTHON · NUMPY

Solve an equation with no data

Train a network to satisfy a differential equation directly, with no solution data at all. The loss is the equation's residual. This is the physics-informed neural network — the method that named the field.

01Challenge

Here is a strange idea that works. Take a network u_θ(t) with random weights. Do not show it a single (t, u) example. Instead, at a scatter of points across the domain, compute the network's own second derivative u''(t) and penalize how much u'' + ω²u differs from zero — that is the harmonic-oscillator equation. Add two more penalties for the initial conditions u(0)=1 and u'(0)=0. Minimize that loss. The Bench does exactly this with a real ferromotion PINN. Press Train and watch the green network curve — driven only by 'obey the equation' — rise out of noise and lock onto the gold cos(ωt). No data. The physics was the supervision.

Predict firstCan a neural network solve a differential equation using ZERO training data — no example solutions at all?

02Model

A physics-informed neural network puts the differential equation INTO the loss. Write the equation as a residual — for u'' + ω²u = 0 the residual is f(t) = u''(t) + ω²u(t) — and drive f to zero at randomly sampled 'collocation' points, adding terms for the boundary or initial conditions. Because there is no data term inside the domain, the network is supervised entirely by physics; it interpolates the true solution between the conditions. The technical key, and the reason this was new: the loss needs the network's OWN derivatives with respect to its input, u'(t) and u''(t), used as outputs. ferromotion computes them exactly by propagating a second-order jet (u, u', u'') through the network on the same autodiff tape that computes the parameter gradient — so the whole thing trains with exact gradients, no finite differences. Two payoffs beyond elegance: a trained PINN evaluates a PDE solution in microseconds where a classical solver takes minutes, and — the version that matters for robotics — you can mix a physics-residual loss where you have no data with a data loss where you do, so the model extrapolates safely into the unmeasured regions.

03Guided practice

Step A (start from noise): the untrained network is nowhere near cos(ωt) — max error above 1. Step B (train on the equation): press Train ×3000. The physics loss falls and the green curve snaps onto the gold cosine; max error drops below 0.01. Step C (appreciate it): nothing ever showed the network the value of cos at any point — it reconstructed the whole solution from 'u'' + ω²u = 0' plus the two initial conditions. Step D (why it matters): the same trick works for PDEs (heat, waves, Navier–Cauchy for soft robots) and, combined with a data loss, lets a model fill gaps where you have physics but no measurements.

04Feedback

PASS WHEN PASS when the network solves u''+ω²u=0 to low error — a solution learned from the equation's residual and the initial conditions alone.

If your run fails, check:

Press Train ×3000 and watch the green network curve lock onto the gold cos(ωt).
If it hasn't converged, train again — the physics loss keeps driving the residual toward zero.

05Retrieve & space

A PINN uses no solution data inside the domain — so what supplies the supervision that shapes u(t)?
Why does the PINN loss need the network's derivative with respect to its input, and how does ferromotion get it exactly?
How does combining a physics-residual loss with a data loss let a model extrapolate safely where measurements are missing?

06Mastery & project

You trained a network to solve a differential equation with zero solution data — the physics-informed neural network. The equation, written as a residual and driven to zero, was the entire teacher. That is physics injected into the loss.

The loss is one of three insertion points. The next module injects physics into the ARCHITECTURE — networks whose very structure conserves energy or respects the Lagrangian — and revisits the energy-drift problem from Module 2.

GO FURTHER · THE LIVE PLAYGROUND

Raissi et al. — the original PINN paper ↗ — physics-informed neural networks for forward and inverse PDE problems — the paper that named the field.

← Energy, and why integrators drift Conserve energy by construction →

PYTHON · NUMPY

Solve an equation with no data

Live ferromotion-wasm: a PINN solving u''+ω²u=0 from the equation alone. Train it and watch the network lock onto cos(ωt) with no solution data.

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