Theoretical physics

AI for a Theoretical Physicist

Use differentiable and data-driven models without discarding dimensional analysis, conservation laws, numerical baselines, or uncertainty.

15-21 hours · estimatedSelf-guided · learn at your pace
Before you start

Physics learners comfortable with differential equations, linear algebra, and Python.

  • Undergraduate classical mechanics
  • Linear algebra and multivariable calculus
  • Python, NumPy, and basic optimization
LESSON 1 · 3-4 hours ESTIMATED

Start from equations and numerical error

Create a trusted numerical baseline and distinguish discretization error from model error.

Scientific machine learning needs a reference problem whose equations and numerical behavior are understood. Use the damped pendulum as a dimensionless initial-value problem and solve it with SciPy’s solve_ivp. Compare RK45 results at several relative and absolute tolerances, then check convergence against a high-accuracy solution. Plot angle, angular velocity, and mechanical energy; with damping, energy should decrease according to the modeled dissipation rather than remain constant. Generate noisy observations only after this baseline is verified. Keep separate variables for physical parameters, solver tolerances, sampling cadence, and observation noise. A dense neural surrogate can appear to fit data while learning the solver’s error, interpolation pattern, or unit convention. Test this by training a small regression model on one cadence and evaluating on another. Report errors in dimensionless and restored physical units. The exercise establishes a hierarchy: equations define the target, numerical analysis bounds the reference, and machine learning is judged against both rather than against a visually smooth curve.

Try it yourself

  1. Nondimensionalize a damped-pendulum equation and solve it with solve_ivp.
  2. Perform a tolerance-convergence check and verify the expected energy trend.
  3. Generate noisy observations and test a simple surrogate across sampling cadences.

You’re ready to move on when…

  • Reference solution converges as tolerances tighten.
  • Energy behavior agrees with the damping term.
  • Surrogate error is compared with numerical and observation error separately.

Learn from the original sources

SciPy: solve_ivp SciPy: least_squares
Enroll above to keep your progress across visits.
PUT IT INTO PRACTICE

Recover and stress-test a damped pendulum

Generate verified synthetic trajectories, recover parameters with autodiff, compare a physics-informed surrogate with solve_ivp, and test shifts in initial condition, cadence, and noise.

What you’ll make

  • Dimensionless derivation
  • Convergence-tested simulator
  • Parameter-recovery notebook
  • Stress-test report

Check your work

  • Numerical reference error is bounded before learning.
  • Recovered parameters include uncertainty and multi-start analysis.
  • Report checks residuals, energy behavior, and out-of-distribution initial conditions.
Project estimate: 6-10 hours beyond lessons. Completion is self-reported; this is not an accredited qualification.

Follow the researchers

Public professional profiles and publications. Contact details don’t imply availability for mentoring.

Miles Cranmer

University of Cambridge

Interpretable machine learning, symbolic regression, and scientific discovery across physics and astronomy.

Jesse Thaler

Massachusetts Institute of Technology

Theoretical particle physics and machine learning for fundamental physics, including collider analysis.