Advanced Topics

The common space, model and observation types connect regression, optimization and optional integrations. Start with normalization and visualization to understand your data and fitted image. Use degree selection, regularization and fixed controls to improve a model under explicit assumptions. Sampling and active learning help decide where to obtain additional observations; assembled domains describe how local charts connect. Objective minimization is a separate path for differentiable costs without regression targets.

Each practical guide explains when the feature is needed, provides runnable examples, and describes how settings affect results. Before/after figures state what changed and retain reproduction scripts and numerical evidence.

Choose a guide by the question you need to answer

Table 2 Practical entry points

Question

Start here

What to inspect afterward

Outputs have very different scales.

Output scaling

Errors in original units

A fit oscillates between measurements.

Cross-validation and degree search, Regularization and numerical evidence

Independent prediction errors

Endpoint or boundary values are exact.

Fixed controls

Full boundary coverage of the constraints

Reference evaluations are expensive.

Explicit ensembles and active learning

Errors on a fixed external validation set

A plot is coarse or misses a feature.

Visualization, Grid sampling

Numerical residuals and render resolution

Independent parameter groups vary together.

Single and assembled domains

Per-factor sums and coordinate order

Neighboring local regions must agree.

Single and assembled domains

Predictions along identified faces

Exact data on a mesh must not overshoot.

Shape-preserving interpolation on simplicial meshes

Prediction range in each cell

A large fit exhausts memory.

Solvers and optional execution

Sparse termination and conditioning

There are costs but no target responses.

Explicit initial models, Objective minimization

Full-sample objective and gradient stopping

Follow a fit through the guides

Start by recording the domain, output units and measurement process. Use sampling and visualization to examine coverage before changing the model. Then choose capacity with held-out data. A larger degree can reduce bias, but increases controls and can worsen conditioning; regularization imposes an additional assumption rather than adding measurements.

Keep the evaluation set fixed while comparing settings. If you change both degree and training locations, a better result cannot be attributed to either change alone. Boundary constraints, coefficient penalties and output transforms should be recorded with every comparison because they change what problem is being solved.

FitResult.success describes numerical termination. Observation rank describes whether data identify the free controls, while held-out errors describe predictive performance. None replaces the others. An accurate interior fit can still miss an important boundary, and a regularized full-rank system can still have an underdetermined data matrix.

The category-theory chapter is a mathematical continuation of the geometric simploid guide. It is useful for understanding faces, degeneracies and gluing abstractly; it is not a prerequisite for fitting a square or prism. The scikit-learn guide uses the same core numerical model in estimator workflows. Exact subdivision in the splitting guide changes charts while preserving the represented function, so assess it differently from refitting.

Guide chapters