Quickstart

This guide gets you up and running with PINA in 5 minutes. By the end, you will have trained a Physics-Informed Neural Network (PINN) to solve the Poisson equation on a unit square.

Install

pip install pina-mathlab

Define a problem

Every PINA workflow starts by defining a SpatialProblem. A problem specifies the output variables, the computational domain, and the conditions (PDE residual, boundary conditions, initial conditions, data) that the solver must satisfy.

from pina import Condition
from pina.problem import SpatialProblem
from pina.domain import CartesianDomain
from pina.equation import Equation, FixedValue
from pina.operator import laplacian

def poisson_equation(input_, output_):
    u = output_.extract(["u"])
    lap_u = laplacian(u, input_, components=["u"], d=["x", "y"])
    return lap_u + (3.14159 ** 2) * u

class PoissonProblem(SpatialProblem):
    output_variables = ["u"]
    spatial_domain = CartesianDomain({"x": [0, 1], "y": [0, 1]})
    domains = {
        "domain": CartesianDomain({"x": [0, 1], "y": [0, 1]}),
        "boundary": CartesianDomain({"x": [0, 1], "y": [0, 1]}),
    }
    conditions = {
        "domain": Condition(
            domain="domain", equation=Equation(poisson_equation)
        ),
        "boundary": Condition(
            domain="boundary", equation=FixedValue(0.0)
        ),
    }

problem = PoissonProblem()
problem.discretise_domain(n=100, mode="grid", domains=["domain", "boundary"])

Create a model

Choose a neural network architecture. For standard PINNs, a FeedForward (MLP) is a solid starting point.

from pina.model import FeedForward

model = FeedForward(
    input_dimensions=2,
    output_dimensions=1,
    inner_size=20,
    n_layers=3,
)

Train with a solver

The PhysicsInformedSingleModelSolver solver wraps the problem and model, and the Trainer orchestrates the training loop.

from pina.solver import PhysicsInformedSingleModelSolver
from pina import Trainer

solver = PhysicsInformedSingleModelSolver(problem=problem, model=model)
trainer = Trainer(solver=solver, max_epochs=1000, accelerator="cpu")
trainer.train()

Inspect results

After training, the model stores its solution in the solver. Evaluate at any point:

import torch

x = torch.tensor([[0.5, 0.5]], requires_grad=True)
u_pred = solver(x)
print(u_pred)

What’s next?

• Walk through the Introductory Tutorial for a deeper introduction.

• Explore the API reference for all available components.

• Read the tutorials for domain-specific guides (Neural Operators, Supervised Learning, etc.).