Simulate and discover

Who this path is for

Scientists and engineers with a domain model to test; calculus is introduced in the shared math lessons.

Preparation

Shared foundations and calculus

Your first useful artifact

A cooling simulator with a convergence table and an independent sensitivity check.

How to study

Work from the cooling model through batched trajectories, sensitivities and parameter fitting. At each step compare against the analytic solution before changing the physical rate or time grid. Use the deeper linear algebra and curvature lessons when parameters are hard to identify. Finish the inverse-problem project, then attempt the two-compartment assessment with its independent conservation check.

Course phases and lessons

1. Setup & first steps

   Run a saved program and explain its environment, array output and device placement. Start on one CPU, practice sharding on four logical CPU devices, and learn how to get a TPU running, launch a TPU job, and run course experiments on TPU in parallel from Phase 00 onward.

   • Set up your learning workspace
   • Meet your arrays and devices
   • Practice with four virtual CPU devices
   • Move your experiment to a TPU
2. Arrays & pure functions

   Treat every axis as part of a contract. Work through normalization, broadcasting and immutable updates, then diagnose mistakes that still return plausible numbers.

   • From NumPy to jax.numpy
   • Shapes, broadcasting, and dtypes
   • Pure functions and explicit inputs
   • Immutable updates and indexing
3. JAX transformations

   Begin with a scalar function, then differentiate it, batch it and compile it. Compare each transformed result with a simple reference so you can explain what changed.

   • Your first gradient
   • Losses and value_and_grad
   • Batch a function with vmap
   • Compile a function with jit
   • Tracing, static arguments, and recompilation
4. State, randomness & control flow

   Make randomness, parameters and loop state explicit. Learn how to repeat an experiment without accidentally reusing randomness or mutating a caller’s state.

   • Random keys without surprises
   • Pytrees and structured parameters
   • Compiled loops with lax.scan
   • Branches with lax.cond
5. Math & optimization

   Connect geometry and probability to the updates in a training loop. Begin with linear algebra, gradient checks and gradient descent, then investigate conditioning, curvature, likelihoods, regularization and optimizer behavior.

   • Vectors, norms, and projections
   • Linear algebra and loss intuition
   • Least squares, rank, and conditioning
   • The chain rule, Jacobians, and directions
   • Gradient checking and numerical accuracy
   • Curvature, Hessians, and learning rates
   • Write gradient descent yourself
   • Probability, likelihood, and stable losses
   • Regularization and honest validation
   • Minibatches, expectation, and gradient noise
   • Optimize with Optax
   • Adam, clipping, and learning-rate schedules
6. Scientific computing

   Build and differentiate a physical simulator, recover a positive rate from observations, and separate numerical accuracy, identifiability and measured execution cost.

   • A functional physical simulation
   • Vectorize trajectories and scan time
   • Differentiate through a solver
   • Fit parameters and scale the simulation

Audit a differentiable scientific inverse problem

A guided project is available for this route.

Complete the staged project contracts and the changed-condition synthesis assessment with independent calculations, observed outputs, and failure diagnosis.

• Convergence against the analytic solution at multiple rates and time grids
• Solver sensitivities checked by independent finite differences
• Parameter recovery, held-out errors and a diagnosis of numerical bias or non-identifiability

Read the portfolio project guide

Work through the synthesis assessment

Use this path in your learning workspace