Technical Labs
Interactive systems for understanding the mathematical structure behind machine learning.
Spectral PDE Lab
Make PDE structure executable.
Browser-side Fourier solvers for the classical PDE archetypes that show up in diffusion models, flow matching, PINNs, and neural operators. Exact under stated assumptions: periodic boundary conditions, smooth initial data, evaluated at FFT-resolved Fourier modes.
Lab page foregrounds the verification scope: periodic boundary conditions, FFT-resolved modes, and exactly which PDE assumptions are being used.
Proof Tasks
Check claims against the source page.
Review-stage multiple-choice checkpoints anchored to real definitions, formulas, and theorems on canonical TheoremPath pages. The first set covers softmax axis choice, attention tensor shapes, reverse-mode backprop, a heat-equation Fourier multiplier, and the SDE/Fokker-Planck distinction.
Noindexed while the wording, notation, and scoring surface are being reviewed.
Attention Mechanics Lab
Shapes, softmax, and Q/K/V attention.
Trace the exact tensor operations that turn dot products into attention weights. Identify which axis softmax normalizes over in attention logits with shape [batch, heads, query_tokens, key_tokens], and verify what breaks if you normalize over heads instead.
Initial coverage is provided by the softmax-axis and attention-shape-trace proof tasks. A dedicated interactive lab with shape diagrams and per-axis simulation is queued for a follow-up.
What these labs are for
Most ML learning tools focus on isolated lessons or coding tasks. TheoremPath focuses on the structure underneath: prerequisites, proof obligations, interactive labs, and diagnostics that show what to study next.
Labs are how that structure becomes inspectable. Spectra, gradients, tensor shapes, solver assumptions, and failure modes are visible objects in a lab, not paragraphs in an article. Each lab states the assumptions under which its computation is exact and links back to the topic page where the underlying theory is developed.