TL;DR
Physics-Informed Neural Networks (PINNs) solve PDE forward and inverse problems by parameterizing the solution u(x,t) with a neural network and enforcing the governing equation via automatic differentiation at collocation points. The framework supports sparse/noisy data and can identify unknown PDE parameters alongside the solution.
Problem
Traditional solvers require meshes and are expensive to couple with parameter inference. PINNs aim to solve (and calibrate) PDE models directly from sparse observations by embedding the PDE as a soft constraint in the training objective.
Benefits vs others
- Mesh-free training using collocation points; handles irregular sensor locations naturally.
- Unified treatment of forward simulation and inverse parameter identification.
- Automatic differentiation provides exact derivatives of the neural solution (up to machine precision).
Interesting detail
- PINNs popularized the idea that PDE constraints can regularize learning from scarce data; later work improved sampling, optimization, and constraints.
- In partial observation settings, PINNs often need careful weighting and adaptive sampling to avoid local minima.
Core method (math)
Template for PINN / physics-constrained. Paper-specific equations are added when manually curated.
\[u_\theta(x,t)\approx u(x,t)\quad\text{(neural approximation)}\]
\[f_\theta(x,t) = \partial_t u_\theta(x,t) + \mathcal{N}[u_\theta(x,t);\lambda]\quad\text{(PDE residual)}\]
\[\mathcal{L} = \mathcal{L}_{\text{data}} + w_r\,\mathcal{L}_{\text{res}} + w_b\,\mathcal{L}_{\text{BC/IC}}\]
\[\mathcal{L}_{\text{res}} = \frac{1}{N_r}\sum_{i=1}^{N_r}\big\|f_\theta(x_i,t_i)\big\|_2^2\]
Main theoretical contribution
- PINNs can be seen as constrained function approximation where the constraint is enforced in expectation over collocation points.
- Inverse problems treat unknown PDE parameters λ as trainable variables optimized jointly with θ.
Main contribution
- Unified forward + inverse PDE solving with a single differentiable model.
- Leverages AD for PDE residuals.
- Catalyzed a large body of physics-informed learning research.
Main results (headline)
(Optional) Add main_results for a quick headline summary.
Experiments
PDE problems
- Burgers
- Navier–Stokes
- Schrödinger
- Wave equation
Tasks
- Forward PDE solve
- Inverse parameter estimation
Experiment setting (high level)
- Collocation points for residual; boundary/initial data; possibly sparse observations.
Comparable baselines
- Numerical solver
- DGM
- Deep Ritz
Main results
Key results
| Task | Metric | Reported takeaway |
|---|---|---|
| Inverse problems | Parameter error | Recovers PDE parameters from sparse/noisy data in reported cases. |
| Forward solve | Solution error | Produces accurate solutions when optimization is well-conditioned. |
Citation (BibTeX)
@article{raissi2019pinns,
title={Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E.},
journal={Journal of Computational Physics},
volume={378},
pages={686--707},
year={2019},
publisher={Elsevier}
}