Physics-Informed Physical AI
The physical world will not be solved by data alone: action-labeled data is scarce, and a black-box net that has never heard of energy or momentum extrapolates dangerously. This course teaches the alternative, inject the physics you already know into a learner, at one of three places: the data (guided), the loss (informed), or the architecture (encoded). You build every method on-device in the browser, driving the real ferromotion Rust library compiled to WebAssembly: reverse-mode autodiff, physics-informed neural nets, Lagrangian and Hamiltonian nets, Neural ODEs, structure-preserving integrators, equation discovery, and differentiable control. Physics → AI → Physical AI, one runnable rung at a time.
▶ Start the course ← All coursesWhere this sits, and what moves it.
Binding constraint · Action-labelled samples per unit of dynamics learned. Physics you already know -- a conservation law, an equation of motion -- is information you do not have to pay for in data, and the exchange rate between the two is what this whole field is trading on.
A learner either knew physics because someone hard-coded it or learned everything from scratch. There was no principled place to put a conservation law you already believed, so it was either an assumption or an absence.
There are three places -- the data, the loss, the architecture -- and this course builds all three plus discovery. the ordering matters: a loss penalty is soft, so a low loss is not a correct answer, which is why the verify-first module exists rather than being an appendix.
The frontier is architectural rather than penalty-based: structures that cannot violate the physics regardless of what the optimiser does, so the guarantee survives training. Where that is hard is exactly where the lumped model breaks -- fluids, contact, deformation -- and that is where the course spends its last modules on purpose.
Every hard thing was impossible until the constraint that made it impossible was named. How we read a frontier →
How machines learn
Before physics, the engine: how a learner turns error into improvement. Build the exact-gradient machine every later method stands on, and see why the physical world resists data-only learning.
- L3The machine that follows gradientsA learner measures how wrong it is, then changes its parameters in the direction that reduces the error fastest. What is that direction?Watch a learner improve by computing the exact gradient of its error and stepping downhill, and meet the engine that makes it exact: reverse-mode automatic differentiation, the keystone under every method in this course.→
- L3The universal approximatorA single neuron can only draw a straight boundary. What does stacking them into layers buy you?Stack neurons into a network and watch it bend onto any continuous curve. Meet the multilayer perceptron and the theorem that makes it powerful, and the catch that makes physics necessary.→
Physics as prior
The knowledge a black box lacks. Meet the equation of motion and the energy structure of a physical system, the priors the rest of the course injects, and the three places they can enter a learner: the data, the loss, or the architecture.
- L3The equation of motionAny rigid mechanical system obeys M(q)q̈ + C(q,q̇)q̇ + G(q) = τ. How much of that did the network in the last lesson know?See the physics a black box has never heard of: the equation of motion M(q)q̈ + C(q,q̇)q̇ + G(q) = τ, computed live by ferromotion as a double pendulum swings. This structured knowledge is the prior every method from here on injects into a learner.→
- L3Energy, and why integrators driftYou simulate a FRICTIONLESS pendulum with a simple explicit-Euler integrator and let it swing for a long time. What happens to its total energy?A frictionless system conserves energy forever, but a careless simulator invents or leaks it. See why the update rule decides, meet structure-preserving integration, and understand the failure mode that haunts every learned dynamics model.→
Physics in the loss
The first place to inject physics: the loss function. Instead of showing a network the answer, penalize it for violating the governing equation, and it solves the equation from the physics alone.
Physics in the architecture
The third and deepest place to inject physics: the network's own structure. Build models that conserve energy or respect the equations of motion because of how they are wired, not because a loss asked them to.
- L3Conserve energy by constructionA black-box network is trained to predict a frictionless orbit, then rolled out for a long time. What happens to the orbit's energy?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.→
- L3Learn the structure, not the mapA Deep Lagrangian Network refuses to learn the whole map from state to torque. What does it learn instead?Build a network that keeps the equation of motion's structure and learns only its physical pieces, the mass matrix and the potential, with the inertia guaranteed valid by construction. This is the Deep Lagrangian Network, and it recovers ferromotion's own dynamics from data.→
- L3Learn the flow, predict the futureEvery method so far predicted a value. A Neural ODE predicts something else. What?Learn a system's continuous dynamics by training THROUGH an ODE solver, the Neural ODE, and watch its rollout predict the trajectory past the data it was trained on. Meet the integrator-in-the-loop idea behind Variational Integrator Networks.→
Discovery
Turn the problem around: instead of assuming the equations and learning their pieces, discover the equations themselves from data, as a sparse formula you can read, or as a coordinate change that makes nonlinear dynamics linear.
- L3Discover the equationYou have only noisy position data from a swinging pendulum and no equations. Can an algorithm recover the actual governing law from the data alone?Recover a system's governing equation directly from data, not a black-box fit, but an actual formula. Meet SINDy and the sparsity knob that separates the real terms from the noise.→
- L3Make it linearA linear model predicts forever and unlocks the whole linear control toolbox. Most systems are nonlinear. What does the Koopman operator do about it?Turn a nonlinear system into a linear one, not by approximating, but by changing coordinates. Meet the Koopman operator: lift the state through the right observables and the dynamics become exactly linear.→
- L3Interpretable by constructionAsk a black-box network what the slope is near a given point. What can it tell you?Build a model whose every part is a number you can read, a blend of local linear models, and get an accurate, data-efficient fit that an engineer can inspect and trust, unlike a black-box network.→
Physical AI: closing the loop
Put the models to work. A model you can differentiate is a model you can control by gradient descent, so tune a controller by backpropagating through the loop, then close the whole arc: learn a model from data and train a controller inside it.
- L3Control by gradientYou have spent the course building models you can differentiate. What does that buy when you close the loop?Tune a controller by backpropagating a trajectory cost through the closed loop, the payoff of a differentiable model. See why one gradient per rollout beats reinforcement learning's thousands of episodes.→
- L3Learn in imaginationYou are handed an unknown plant. You identify a model from data, tune a controller entirely inside that model, then deploy. Where does the risk sit?Close the whole loop: identify a model of an unknown plant from data, tune a controller entirely inside that learned model, and deploy it on the real system. The complete physics-informed pipeline, on-device.→
- L3The industry agrees: structure-as-differentiation, at scaleA world model predicts the next video frame directly, pixel by pixel, from the current frame and an action, no explicit position, velocity, or mass anywhere inside it. Can it violate conservation of momentum during a predicted collision?See that the principle this course has built lesson by lesson (a physics-structured, differentiable model, identified and controlled by gradient) is the same bet three 2025–2026 efforts are placing at industrial and research scale: Newton (GPU differentiable physics, contributed to the Linux Foundation by Disney Research, Google DeepMind, and NVIDIA), gradient-based system identification on real robots using MJX, and OrbiSim's argument that a world model should itself be a differentiable physics engine rather than a generative video predictor.→
The on-device fabric
A learned model is only as good as the silicon it runs on. Your device is a heterogeneous compute fabric (CPU, GPU, NPU) and smart software routes each workload to whatever wins. See it happen, live, on your own hardware.
- L3Your device is a fabricEvery model in this course trained on-device, in your browser. On a modern machine, which backend should each workload run on?See the real cross-fabric runtime measure your device and route each workload to the backend that wins on it (tiny work to the CPU, large work to your GPU) the adaptive heterogeneous compute the whole course runs on.→
- L3"It Runs" Is Not "It Runs In Time"Your policy is correct and every test passes. It now has to run at a hundred hertz on a robot. Which of these will a passing test never show you?Measure three costs that a passing test cannot see: whether the policy meets its deadline, whether the control tick allocates, and whether your own benchmark number is allowed to be quoted. Pass = you drive the clock into each of its three health states, explain why two of them are different failures rather than degrees of one, and make a timing measurement refuse to be quoted.→
The model meets reality
Close the loop with the physical world: calibrate the model from recorded data by exact gradients through the dynamics, and find out what the numbers you grade a grasp with leave out.
- L3The model meets realityEvery model so far was right by construction: you knew the equations and the parameters. A real robot hands you neither. What do you do?Calibrate a robot's masses and friction from recorded torques by exact gradients through the dynamics, real-to-sim, the single most validated use of differentiable physics.→
- L4The Grasp Number Lies TwiceYou certify a grasp with a force-closure quality and a stick-or-slip flag. Both come back fine. What have you actually established?Take the two numbers everyone grades a grasp with - a force-closure quality and a stick-or-slip flag - and find what each one hides. You will discover that a coplanar grasp is full rank in six dimensions rather than degenerate; that the force-closure number is a direction-sampled UPPER bound whose value falls threefold as you probe more finely, so a claim read off it can be a property of the sampler rather than of the grasp; and that a contact reported as stuck has usually already lost a quarter of its stuck area. Pass = you read the rank, read how far the quality metric still moves under finer sampling, and find the load at which a stick-area monitor warns against the load at which a Coulomb monitor does.→
- L3The Robot Is the FileSomeone hands you a robot description. It parses, the articulation moves, every number is plausible, nothing warns. Where is the bug?Price three conventions in a robot description in newton-metres, and find out that the pose you test in decides which of them you can see. You will read a joint limit that means something different depending on the joint type, watch a length-scale error change a torque by a factor that is not the length ratio, flip a header token and move gravity, and discover a pose where a hundred-fold error is exactly invisible. Pass = you state all three errors as numbers and explain why one of them cannot be undone by rescaling the answer.→
Fluids: the physics you can't lump
Know when the lumped coefficient model of a rotor, hull, or pour breaks, and how a resolved, verified, differentiable fluid solver reveals and repairs it, live in the browser.
Verify-first: don't trust, certify
A physics-informed net only SOFTLY penalizes the physics, so a low loss is not a correct answer. Turn a trained solution, and a trained policy, into ones you can trust by computing a certificate on the output: the Institute's verify-first spine, from a PINN's residual bound to an energy that certifies every action.
- L4Don't trust a PINN, certify itA physics-informed net's training loss went to nearly zero. Does that guarantee its solution is correct?A PINN drives a residual toward zero, a soft penalty, not a guarantee. Compute an a-posteriori certificate: a rigorous bound on the true error from the trained net's own residual, needing no true solution. Then watch the same idea become an energy certificate on a controller and on a frozen policy. Soft → structural → proven.→
- L4The Gradient That LiesYou take a contact you can solve exactly and refine the simulator's timestep. What happens to the gradient it returns?Measure, on a contact you can solve exactly, that the gradient a differentiable simulator returns does not converge as the contact stiffens - it diverges as the square root of stiffness and points the wrong way over most of the usable range. Then find out WHICH PART of the simulator is responsible, because the obvious answer is the wrong one. Pass = you read the divergence exponent off the lab, find a stiffness where the sign is wrong, and show the same contact model getting the right answer once the integrator changes.→
- L4The Reference That Shares Its MachineryYou want to split the contact-gradient error into its causes. Your reference solver shares the same integrator. What does the comparison measure?Take a measurement you already trust and find out what it actually measures. You will split the contact-gradient error into the part the integrator causes and the part the contact model causes, discover they move in opposite directions, and learn the two ways a reference lies to you: by sharing machinery with the thing it scores, and by having a noise floor above the number you wanted to quote. Pass = you name the integrator share and the model share at 1e6, and say why an answer that moves when you change the probe is not an answer.→
- L4What a Certificate Cannot ClaimA verification returns UNDECIDED. What does that tell you?Learn to read a verification result as a claim with preconditions rather than a number. You will make the same system return Certified, Refuted and Undecided; discover that one of those transitions involves no change to the geometry at all; find a green verdict that means nothing; and find a fourth failure the lab uncovered in itself, a bound that was only valid in a region nobody checked. Pass = you produce all three verdicts, explain which control caused each, and say what the precondition check found.→