pina-multiphysics

name	pina-multiphysics
description	Compose coupled multi-field PINA problems (e.g. thermo-elasticity, magnetohydrodynamics) by listing multiple EquationSpecs on one ProblemSpec — no new schema needed.
triggers	["multiphysics","coupled pde","thermo-elastic","magnetohydrodynamics","mhd","reaction-diffusion","navier-stokes thermal","phase field"]

PINA Multiphysics Composition

Use this skill when the user's problem has more than one PDE coupled through shared fields. Typical examples:

• Thermo-elasticity — heat equation + elasticity + thermal-strain source term.
• Reaction-diffusion — two species coupled via non-linear reaction.
• Navier-Stokes with heat transfer — momentum + continuity + heat equation with buoyancy.
• MHD — Navier-Stokes + Maxwell with Lorentz-force coupling.

Key insight: no new schema

The existing ProblemSpec already supports it. Just list multiple EquationSpec entries that share output_variables and point each one at a distinct interior subdomain. PINA emits one loss term per condition; the shared output network couples them through gradient descent.

Recipe

1. In output_variables list every field that any equation references: e.g. ["T", "ux", "uy"] for 2D thermo-elasticity.

2. Create one interior subdomain per equation — PINA keys conditions on subdomain names, so you cannot have two conditions on the same name. Keep the actual bounds identical; only the names differ:

   SubdomainSpec(name="D_heat",    bounds={"x": [0,1], "y": [0,1]}),
   SubdomainSpec(name="D_elastic", bounds={"x": [0,1], "y": [0,1]}),
   

3. Declare each PDE as its own EquationSpec. List in outputs only the fields that appear in the coupling term of that equation — the composer only extracts those from output_.

4. Attach each equation to its subdomain with one ConditionSpec(kind="equation", equation_name=...).

5. Boundary / initial conditions stay as before; reuse the same wall subdomains for every field (FixedValue applies to each declared output component automatically).

Worked example — 2D thermo-elasticity

from marimo_flow.agents.schemas import (
    ConditionSpec, DerivativeSpec, EquationSpec,
    ProblemSpec, SubdomainSpec,
)

spec = ProblemSpec(
    name="thermoelastic_plate",
    output_variables=["T", "ux", "uy"],
    domain_bounds={"x": [0.0, 1.0], "y": [0.0, 1.0]},
    subdomains=[
        SubdomainSpec(name="bottom", bounds={"x": [0,1], "y": 0.0}),
        SubdomainSpec(name="top",    bounds={"x": [0,1], "y": 1.0}),
        SubdomainSpec(name="left",   bounds={"x": 0.0,   "y": [0,1]}),
        SubdomainSpec(name="right",  bounds={"x": 1.0,   "y": [0,1]}),
        SubdomainSpec(name="D_heat",    bounds={"x": [0,1], "y": [0,1]}),
        SubdomainSpec(name="D_elastic", bounds={"x": [0,1], "y": [0,1]}),
    ],
    equations=[
        EquationSpec(
            name="heat",
            form="T_xx + T_yy",
            outputs=["T"],
            derivatives=[
                DerivativeSpec(name="T_xx", field="T", wrt=["x","x"]),
                DerivativeSpec(name="T_yy", field="T", wrt=["y","y"]),
            ],
        ),
        EquationSpec(
            name="elastic_x",
            form="ux_xx + ux_yy - alpha*T",
            outputs=["ux", "T"],                 # COUPLING — T appears here
            derivatives=[
                DerivativeSpec(name="ux_xx", field="ux", wrt=["x","x"]),
                DerivativeSpec(name="ux_yy", field="ux", wrt=["y","y"]),
            ],
            parameters={"alpha": 0.5},
        ),
    ],
    conditions=[
        ConditionSpec(subdomain="bottom", kind="fixed_value", value=0.0),
        ConditionSpec(subdomain="top",    kind="fixed_value", value=0.0),
        ConditionSpec(subdomain="left",   kind="fixed_value", value=0.0),
        ConditionSpec(subdomain="right",  kind="fixed_value", value=0.0),
        ConditionSpec(subdomain="D_heat",    kind="equation", equation_name="heat"),
        ConditionSpec(subdomain="D_elastic", kind="equation", equation_name="elastic_x"),
    ],
)
cls = compose_problem(spec)

Choosing the network architecture

Pick a single feed-forward / FNO / DeepONet with output dim = len(output_variables) so the fields share representation capacity. The Model agent does this automatically when it sees multiple outputs.

Validation expectations

• Each equation gets its own loss term in the trainer metrics (D_heat_loss, D_elastic_loss, …).
• If one loss term plateaus while another keeps improving, add loss-weights via SolverPlan.weights (not yet wired in Phase A-0 — escalate if needed).

When to split into two problems instead

Couple via forward time stepping if the physics are weakly coupled and run at different timescales (e.g. fast chemistry + slow flow). Multiphysics composition is ideal for tightly coupled steady-state or homogeneous-timescale problems.