Forward vs Inverse Problems

AnyPINN supports both forward and inverse problems. This guide explains the difference and how the library handles each.

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What's the difference?

Forward problem: All parameters are known. The PINN approximates the solution of the differential equation.

\[ \text{Given } \frac{dy}{dt} = f(t, y, \theta) \text{ with known } \theta, \text{ find } y(t). \]

Inverse problem: Some parameters are unknown. The PINN simultaneously approximates the solution and recovers the parameters from partial observations.

\[ \text{Given } \frac{dy}{dt} = f(t, y, \theta) \text{ and observations } y_i^{\text{obs}}, \text{ find } y(t) \text{ and } \theta. \]

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Inverse problems in AnyPINN

This is AnyPINN's primary use case. The key abstraction is the Argument / Parameter split:

# Everything the equation needs is in args
props = ODEProperties(
    ode=my_ode,
    y0=torch.tensor([...]),
    args={
        "gamma": Argument(0.1),  # Known, fixed value
    },
)

# Unknown quantities go in params
params = ParamsRegistry({
    "beta": Parameter(config=hp.params_config),  # Unknown, learned
})

The callable accesses both the same way:

def my_ode(x, y, args):
    beta = args["beta"](x)   # Works for both Argument and Parameter
    gamma = args["gamma"](x)
    ...

The ODEInverseProblem class composes three constraints:

1. Residual loss: enforces the differential equation on collocation points
2. Initial condition loss: enforces the initial state
3. Data loss: fits the predicted fields to observations

All three are necessary for a well-posed inverse problem.

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Forward problems in AnyPINN

For forward problems, there are no unknown parameters and typically no observed data to fit against. The loss has only two terms:

1. Residual loss: enforces the differential equation
2. Initial/boundary condition loss: enforces constraints

AnyPINN supports forward problems through two paths:

ODE forward problems

Use ODEInverseProblem with no parameters and no data constraint:

props = ODEProperties(
    ode=my_ode,
    y0=torch.tensor([...]),
    args={
        "beta": Argument(0.3),
        "gamma": Argument(0.1),
    },
)

problem = ODEInverseProblem(
    props=props,
    hp=hp,
    fields=fields,
    params=ParamsRegistry({}),  # No learnable parameters
)

PDE forward problems

Use the PDE constraint classes directly for more control over boundary conditions:

from anypinn.problems import (
    PDEResidualConstraint,
    DirichletBCConstraint,
    BoundaryCondition,
)

See PDE Forward Problems for a full walkthrough.

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When to use which

Scenario	Type	Example
Recover rate constants from epidemic data	Inverse	SIR model
Solve a heat equation with known conductivity	Forward	Heat 1D
Estimate diffusivity from temperature measurements	Inverse	Inverse Diffusivity
Compute the steady state of a reaction-diffusion system	Forward	Allen-Cahn

Most of the built-in catalog templates are inverse problems. The PDE templates (Poisson 2D, Allen-Cahn) demonstrate forward problems.