define-equations

# Define Equations 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 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: ```python FixedGradient(0.0, components=["theta"], d=["t"]) ``` ## Step 4 — Custom equation functions Import the required utilities: ```python from pina.operator import grad from pina.equation import Equation ``` ### Standard PDE (2 arguments) ```python 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): ```python 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. ```python 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