Scientific computing · Machine learning

What Are Physics-Informed Neural Networks (PINNs)?

July 14, 2026 · ← All posts

Most machine-learning models learn from data alone: show a network enough examples and it learns to interpolate between them. That works well when data is cheap. In science and engineering, it usually isn't — experiments are expensive, sensors are sparse, and simulations take hours on a cluster. But we have something most ML problems don't: we already know the laws the answer must obey.

A Physics-Informed Neural Network (PINN) is a neural network that is trained to satisfy those laws — typically a partial differential equation (PDE) — in addition to (or even instead of) fitting measured data. The idea was popularized by Raissi, Perdikaris, and Karniadakis in their 2019 paper, and it has since grown into an active subfield of scientific machine learning.

The core idea

Suppose we want the temperature field, velocity field, or concentration profile u(x, t) of some physical system. Instead of solving the governing PDE with a mesh-based method like finite volumes or finite elements, a PINN represents the solution directly as a neural network: the network takes the coordinates (x, t) as input and returns the predicted value of u at that point.

The trick is in the loss function. Because neural networks are differentiable, we can use automatic differentiation — the same machinery that computes gradients for training — to compute the derivatives of the network's output with respect to its inputs: ∂u/∂t, ∂u/∂x, ∂²u/∂x², and so on. Plugging those derivatives into the governing equation tells us how badly the network violates the physics at any point. That violation, called the residual, becomes part of the training loss:

Loss = L_data + L_physics + L_boundary Fit the measurements, satisfy the PDE at sampled points, and honor initial/boundary conditions.

Concretely, for the viscous Burgers' equation — a classic PINN benchmark —

∂u/∂t + u·∂u/∂x − ν·∂²u/∂x² = 0 The physics loss is the mean squared value of this left-hand side, evaluated at thousands of random points in space and time.

Training then does two things at once: it pulls the network toward the observed data points, and it pushes the residual toward zero everywhere else. The physics acts as a regularizer built from first principles — the network can't invent behavior the PDE forbids, even in regions where there is no data at all.

Why this is attractive

Mesh-free. There is no grid to generate. The PDE is enforced at randomly sampled points, which is appealing for complex geometries and high-dimensional problems where meshing is painful.
Sparse, noisy data is fine. A handful of sensor readings plus the governing equations can reconstruct an entire field. The physics fills in the gaps.
Inverse problems come almost for free. Unknown physical parameters — a viscosity, a diffusion coefficient, a source term — can be made trainable variables and inferred alongside the solution. Doing this with classical solvers usually means wrapping an expensive optimization loop around repeated forward solves.
Differentiable end to end. The trained model gives you gradients of outputs with respect to inputs and parameters, which is useful for design optimization and sensitivity analysis.

Where the honest caveats are

PINNs are not a drop-in replacement for classical solvers, and it is worth being direct about that. For a well-posed forward problem on a reasonable geometry, a mature finite volume or finite element code is typically faster and more accurate than training a neural network from scratch. Training PINNs can also be finicky: stiff PDEs, sharp gradients, shocks, and multi-scale physics are hard for them, balancing the loss terms often requires manual tuning, and networks have a "spectral bias" that makes high-frequency features slow to learn.

A useful rule of thumb: reach for a PINN when the problem is data-plus-physics — inverse problems, assimilation of sparse measurements, unknown coefficients — rather than a pure forward solve that a classical code already handles well.

Where they fit in practice

In an R&D setting, the most compelling uses I see are: inferring hidden fields from limited measurements (e.g., full flow fields from a few probes), estimating material or transport parameters from experiments, building fast differentiable surrogates that stay physically consistent, and handling problems in high dimensions where meshes give up. Ecosystem-wise, libraries like DeepXDE and NVIDIA's PhysicsNeMo (formerly Modulus) make it straightforward to experiment without writing the residual plumbing yourself.

The bigger picture: PINNs are one member of a growing family — neural operators, physics-constrained surrogates, differentiable simulators — that blend three centuries of physical modeling with modern machine learning. The common thread is the one worth remembering: when you know the equations, make the model obey them.