PAI-230 · Education

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.

L3·10 modules · 16 live labs·19 lessons
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Module 1

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.

  1. L3The machine that follows gradientsWatch 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.→
  2. L3The universal approximatorStack 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.→
Module 2

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.

  1. L3The equation of motionSee 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.→
  2. L3Energy, and why integrators driftA 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.→
Module 3

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.

  1. L3Solve an equation with no dataTrain 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.→
Module 4

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.

  1. L3Conserve energy by constructionBuild 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.→
  2. L3Learn the structure, not the mapBuild 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.→
  3. L3Learn the flow, predict the futureLearn 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.→
Module 5

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.

  1. L3Discover the equationRecover 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.→
  2. L3Make it linearTurn 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.→
  3. L3Interpretable by constructionBuild 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.→
Module 6

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.

  1. L3Control by gradientTune 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.→
  2. L3Learn in imaginationClose 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.→
  3. L3The industry agrees: structure-as-differentiation, at scaleSee 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.→
Module 7

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.

  1. L3Your device is a fabricSee 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.→
Module 8

The model meets reality

Close the loop with the physical world: calibrate the model from recorded data by exact gradients through the dynamics.

  1. advancedThe model meets realityCalibrate 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.→
Module 9

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.

  1. advancedWhen coefficients breakKnow when the lumped coefficient model of a rotor, hull, or pour stops telling the truth — and how a resolved, verified, differentiable fluid solver reveals the fix, live in your browser.→
Module 10

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.

  1. L4Don't trust a PINN — certify itA 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.→
  2. L4The Gradient That LiesMeasure, on a contact you can solve exactly, that a differentiable simulator's contact gradient 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. Pass = you read the divergence exponent off the lab and find a stiffness where the sign is wrong.→