Define Domains

Define Domains for a PINA Problem

[!IMPORTANT] Read 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

from pina.domain import CartesianDomain, EllipsoidDomain, SimplexDomain

CartesianDomain supports sampling modes: random, grid, chebyshev, latin/lh.

Set operations on domains

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:

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:

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:

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:

problem.discretise_domain(n=5000, mode="random", domains=["D"])
problem.discretise_domain(n=500, mode="random", domains=["boundary"])

After all domains are discretised:

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