define-domains

# Define Domains for a PINA Problem > [!IMPORTANT] > Read [RULES.md](../RULES.md) before using this skill — it applies to all skills. > This is a sub-skill of **create-problem**. Load the entry-point skill first. Use this skill to create spatial, temporal, and parameter domains, and to discretise them for training. ## Step 1 — Create domains For each domain type, ask: > What are the variable names and their ranges? If the user does not specify a domain, ask for: 1. Variable name (e.g. `x`) 2. Lower bound (e.g. `0`) 3. Upper bound (e.g. `1`) ### Domain types available ```python from pina.domain import CartesianDomain, EllipsoidDomain, SimplexDomain ``` | Domain type | Description | Example | |--------------------|--------------------------------------|-------------------------------------------| | `CartesianDomain` | Hyperrectangle (most common) | `CartesianDomain({"x": [0, 1], "y": [0, 1]})` | | `EllipsoidDomain` | Hyperellipsoid | `EllipsoidDomain({"x": [0, 1], "y": [0, 1]})` | | `SimplexDomain` | Simplex defined by vertices | `SimplexDomain(vertices=[...])` | `CartesianDomain` supports sampling modes: `random`, `grid`, `chebyshev`, `latin`/`lh`. ### Set operations on domains ```python from pina.domain import Union, Intersection, Difference, Exclusion combined = Union(domain_a, domain_b) overlap = Intersection(domain_a, domain_b) subtracted = Difference(domain_a, domain_b) excluded = Exclusion(domain_a, domain_b) ``` ## Step 2 — Domain methods for problem setup ### `partial()` — extract boundary Creates a sub-domain representing the boundary of the parent domain: ```python spatial_domain = CartesianDomain({"x": [0, 1], "y": [0, 1]}) boundary = spatial_domain.partial() # boundary of the square ``` ### `update()` — combine domains (space + time) Creates the Cartesian product of two domains. Essential for space-time problems: ```python spatial_domain = CartesianDomain({"x": [-1, 1]}) temporal_domain = CartesianDomain({"t": [0, 1]}) interior = spatial_domain.update(temporal_domain) # Equivalent to CartesianDomain({"x": [-1, 1], "t": [0, 1]}) ``` Common patterns: ```python domains = { "D": spatial_domain.update(temporal_domain), # space-time interior "ic": spatial_domain.update(CartesianDomain({"t": 0})), # initial condition "boundary": spatial_domain.partial().update(temporal_domain), # moving boundary } ``` ## Step 3 — Discretise domains (sampling) After the problem class is fully defined, sample points from each domain: ```python problem.discretise_domain(n=5000, mode="random", domains=["D"]) problem.discretise_domain(n=500, mode="random", domains=["boundary"]) ``` | Mode | Description | |---------------|---------------------------------------| | `"random"` | Uniform random sampling (default) | | `"latin"`/`"lh"` | Latin hypercube sampling | | `"grid"` | Regular grid points | | `"chebyshev"` | Chebyshev nodes (good for polynomials)| After all domains are discretised: ```python problem.move_discretisation_into_conditions() assert problem.are_all_domains_discretised ``` ## Checklist - [ ] All required domains are defined (`spatial_domain`, `temporal_domain`, `parameter_domain`, or `unknown_parameter_domain` as appropriate) - [ ] `domains` dict contains a key for every domain referenced in conditions - [ ] For space-time problems: `spatial_domain.update(temporal_domain)` used to build the interior domain - [ ] For boundary conditions: `spatial_domain.partial()` used correctly - [ ] `problem.discretise_domain(n=..., mode=..., domains=...)` called for each physics domain - [ ] `problem.move_discretisation_into_conditions()` called before training - [ ] `problem.are_all_domains_discretised` is `True` after discretisation