define-equations

name	define-equations
description	Define PDEs, ODEs, boundary conditions, and custom equations for PINA physics-driven problems. Covers the equation zoo, custom equation functions, and inverse problem equation signatures.
license	MIT
compatibility	opencode, codex, claude
metadata	{"audience":"users","workflow":"problem-creation"}

Define Equations 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 when the problem is physics-driven and you need to define the governing equations and boundary conditions.

Step 1 — Check the equation zoo

What physical equations apply?

First check if the equation is available in the built-in zoo:

Zoo class	Equation
FixedValue(value)	u - v = 0 (Dirichlet BC)
FixedGradient(value)	∂u/∂n - v = 0 (Neumann BC)
FixedFlux(value)	div(u) - v = 0 (flux BC)
FixedLaplacian(value)	Δu - v = 0 (Laplacian BC)
PoissonEquation(forcing_f)	Δu = f
BurgersEquation(nu)	u_t + u u_x = ν u_xx
AdvectionEquation(beta)	u_t + β·∇u = 0
AllenCahnEquation(ε, α)	u_t = ε u_xx + α (u - u³)
DiffusionReactionEquation(λ)	u_t = λ Δu + f
HelmholtzEquation(k)	Δu + k² u = f
AcousticWaveEquation(c)	u_tt = c² Δu

If the equation matches one of these, use it directly.

Step 2 — Unknown equation

If the equation is not in the zoo:

1. Check if the equation is present in .opencode/skills/define-equations/famous_odes_pdes.json, if not present search the web for the equation form, common PINN implementation, and known boundary/initial conditions.
2. Present the found formulation to the user and ask:

   I found this equation: <formulation>. Should I use this or would you like to provide your own?

3. If the user provides their own, use that instead.

Step 3 — Define boundary conditions

What boundary conditions apply? (Dirichlet, Neumann, Robin, periodic, etc.)

A well-posed PDE problem needs both the PDE and its boundary/initial conditions. Common types:

• Dirichlet: FixedValue(value) — imposes u = value on the boundary
• Neumann: FixedGradient(value) — imposes ∂u/∂n = value
• Flux: FixedFlux(value) — imposes div(u) = value
• Laplacian: FixedLaplacian(value) — imposes Δu = value

For zoo boundary conditions, use components and d parameters when the problem has multiple output variables:

FixedGradient(0.0, components=["theta"], d=["t"])

Step 4 — Custom equation functions

Import the required utilities:

from pina.operator import grad
from pina.equation import Equation

Standard PDE (2 arguments)

def my_pde(input_, output_):
    u_x = grad(output_, input_, components=["u"], d=["x"])
    u = output_.extract(["u"])
    return u_x - u

Inverse problem (3 arguments)

When the problem is also an InverseProblem, the equation function receives a third argument params_ (a dict of unknown parameters):

def my_pde_inverse(input_, output_, params_):
    f = torch.exp(-2 * (input_["x"] - params_["mu1"])**2)
    return laplacian(output_, input_, components=["u"], d=["x"]) - f

Step 5 — Build conditions

Wrap equations in Condition objects. Return to the condition-setup or create-problem skill for this step.

from pina import Condition

conditions = {
    "interior": Condition(domain="D", equation=Equation(my_pde)),
    "dirichlet_bc": Condition(domain="boundary", equation=FixedValue(0.0)),
    "neumann_bc": Condition(domain="boundary", equation=FixedGradient(0.0)),
    "ic": Condition(domain="t0", equation=Equation(initial_cond)),
}

Checklist

• Equation identified (zoo class or custom Equation)
• If equation was unknown: searched the web, presented the found formulation to the user, and confirmed before using
• Both the PDE and boundary/initial conditions are defined
• Custom equations use correct function signature: (input_, output_) for standard, (input_, output_, params_) for inverse
• Operator calls use LabelTensor with correct components and d parameter names