pinnrl

Physics-informed neural networks for forward and inverse PDE problems, evaluated against numerical references.

A physics-informed neural network (PINN) represents the solution u(x, t) of a PDE as a neural network. It is trained to make the PDE residual, the boundary conditions and the initial condition vanish at sampled collocation points, optionally together with measured data. No mesh is needed, and the PDE's parameters can be learned alongside the solution.

pinnrl packages that workflow:

• ten PDE configurations: heat (1-D and 2-D), wave, Burgers, convection, KdV, Allen–Cahn, Cahn–Hilliard, Black–Scholes and a damped pendulum. Each exact solution is checked against its PDE, initial condition and boundary conditions by the test suite.
• seven network architectures;
• adaptive collocation sampling;
• adaptive loss weighting;
• inverse problems (parameter identification);
• data-driven training on The Well datasets;
• a Dash dashboard for launching and inspecting runs.

pip install pinnrl
pinnrl-train --pde burgers --arch fourier --epochs 2000
pinnrl-dashboard

When to use a PINN

Be clear about the trade-off. For a well-posed forward problem in one to three dimensions, classical solvers (finite differences, finite elements, spectral methods) are faster and orders of magnitude more accurate. For the 1-D periodic heat equation with \(\alpha = 0.1\) and \(u_0 = \sin(2\pi x)\), an explicit finite-difference scheme on 100 cells reaches a relative L2 error of about \(6 \times 10^{-4}\) in roughly 30 ms of CPU time; a PINN needs minutes of training to reach about \(10^{-2}\).

PINNs earn their place elsewhere:

Situation	Why a PINN helps
Inverse problems	Unknown coefficients or fields are learned jointly with the solution from sparse, noisy measurements. There is no adjoint code to write.
Physics plus data	Measurements and conservation laws enter the same loss, so gaps in the data are filled by the physics.
Awkward geometry or many dimensions	It is mesh-free: points are sampled, not meshed.
Differentiable surrogates	The trained network is a smooth function of (x, t) that is cheap to evaluate and differentiate.

The long-term goal of the project is the first row applied to industry: digital twins that estimate unmeasured states of real process equipment. See Applications and the Research program.

Where to go next

• Getting started: install, your first run, and the Python API.
• Concepts: how a PINN is trained, and what each technique in the package is for.
• Equations: the ten PDE configurations, where they arise, and why each is hard for a PINN.
• Guides: Training, Inverse problems, Datasets, Dashboard, Benchmarks.

Origins

pinnrl grew out of the master's thesis On Solving Partial Differential Equations (PDEs) Using Physics-Informed Neural Networks (PINNs) (J. García Ventura, Master in Mathematical Modeling, Research, Statistics, and Computation, University of the Basque Country UPV/EHU, 2025; advisors C. M. Cuesta Romero and F. de la Hoz Méndez). Read the thesis.

@software{garcia_pinnrl,
  author = {Garcia Ventura, Jose},
  title  = {pinnrl: physics-informed neural networks with adaptive collocation sampling},
  url    = {https://github.com/josegarciav/PINNs-RL-PDE}
}