Must a PINN use partial differential equations?

Those are a prominent application, but related formulations can incorporate ordinary differential equations and other scientific constraints.

Does it need a traditional mesh?

Many formulations use sampled collocation points without a conventional solver mesh, but sampling and coverage still require care.

Can it discover an unknown physical law automatically?

Some inverse setups estimate specified unknown parameters. That is narrower than unrestricted discovery of the correct governing law.

AI Glossary / Deep learning / Advanced

Physics-informed neural networks

Definition

Physics-informed neural networks, or PINNs, are neural models trained using physical equations or related scientific constraints as well as available observations or boundary information. They commonly approximate solutions to differential equations or estimate unknown parameters in those equations.

What it means

A conventional data-fitting model mainly learns from examples of inputs and outputs. A PINN can also be penalized when its predictions violate a specified governing equation. This gives the learning process information between observed measurements.

For example, a temperature model might combine sensor readings with a heat-transfer equation and conditions at the boundary of the region. The model's usefulness depends on whether those equations and conditions appropriately describe the real problem.

Physics-informed does not mean physically guaranteed. Many PINNs impose equations through a training penalty rather than an exact constraint. Nor does the term mean that the computation happens in special physical hardware. A PINN can run on an ordinary computer and still incorporate scientific knowledge.

How it works

In a common continuous formulation, a neural network maps coordinates such as position and time to a predicted field. Differentiation through the network provides derivatives needed by the governing equation. Substituting the prediction and its derivatives into that equation gives a residual: a measure of mismatch.

Training combines residual penalties at selected collocation points with losses for observations and initial or boundary conditions. The relative weighting and scaling of these terms affect optimization. Some formulations enforce certain conditions by construction instead of penalizing them.

A forward problem seeks the field when the equation and its parameters are given. An inverse problem may learn unknown equation parameters alongside the field from measurements.

Small training residuals at sampled points do not alone establish accuracy throughout the domain. Research has documented optimization failures even where the network can represent a suitable solution. Validation should check independent points, boundary behavior and an appropriate numerical or experimental reference.

Three practical examples

A heat-transfer experiment combines sparse temperature observations with a specified diffusion model to estimate a field.
A parameter-estimation study uses measured trajectories and a known equation form to infer an unknown coefficient.
A scientific surrogate project evaluates whether an equation-informed approximation is useful for a narrowly defined range of conditions.

Think of it this way

Think of completing a puzzle using both visible pieces and rules about how neighboring pieces must fit. The rules supply information where some pieces are missing, but a wrong rule can lead to a coherent-looking yet incorrect result.

The analogy does not imply that training finds the unique correct answer. The equations may be incomplete, measurements noisy and the optimization difficult. Boundary information also matters: the same local rule can allow many different overall solutions.

Worked example

Why the equation and initial condition both matter

Consider the teaching problem u'(t) + u(t) = 0 with u(0) = 1. Define the equation residual r(t) = u'(t) + u(t).

Candidate A is u(t) = 0. Its residual is zero everywhere, but it violates the initial condition: the squared initial error is (0 − 1)² = 1. Candidate B is u(t) = exp(−2t). It satisfies the initial condition, but its residual is −exp(−2t), so at t = 0 the squared residual is 1.

Candidate C is exp(−t), which satisfies both requirements. These hand-written functions illustrate the quantities a PINN training objective can consider; they are not outputs of a trained network. A loss that checks only the equation could accept the wrong zero solution. Checking only the initial value would not determine the correct time evolution.

Illustrative case study

Fictional example: Northbank Thermal develops a PINN to estimate temperatures between a few sensors in a test enclosure. The initial fit closely matches the observed points and has a low average residual at its chosen collocation locations.

Engineers compare the field with additional measurements and a conventional solver. They discover a local region near a boundary where the prediction is poor. The simplified boundary condition does not match the experiment, and the collocation distribution gives that region little attention.

The team corrects the problem specification, revises the sampling strategy and repeats independent validation. It also records the conditions under which the approximation is intended to be used.

The lesson is that incorporating an equation does not remove modeling judgment. Data fit, equation consistency, boundary accuracy and real-world validity are separate pieces of evidence.

Common misconceptions and mistakes

A physics penalty makes a prediction exact. Soft constraints can remain violated after training.
No measured data means no information is needed. Governing equations, domains and initial or boundary conditions still define the problem.
Low average residual guarantees low solution error everywhere. Unsampled regions and difficult dynamics can remain problematic.
PINNs always outperform conventional solvers. Suitability and cost depend on the specific forward or inverse problem.

Business use

PINNs are relevant when scientific structure can help connect limited observations to a model of an underlying process. An organization should first identify whether the task is estimating a field, inferring parameters or building a reusable approximation.

Compare the approach with established numerical methods and other statistical models using the same information and accuracy requirements. Include training expense, sensitivity to assumptions and the cost of changing geometry or boundary conditions. For operational use, define an independent validation standard instead of relying solely on the training loss or on the reassuring presence of a physical equation.

Check your understanding

A model satisfies a differential equation but violates the required initial condition. Has it solved the stated initial-value problem?

Answer: No. Both the equation and the initial condition are part of the problem. Equation agreement alone can admit a different solution.

Related terms

Inverse problem
Automatic differentiation
Collocation (remote sensing)
Neural operators
Scientific machine learning
differential equation
collocation
boundary condition
neural operator

Learn next

Read inverse problem and automatic differentiation. Write down a simple equation, its required conditions and a separate validation measurement, then identify which evidence each term in a proposed training loss supplies.

Sources

Raissi, Perdikaris and Karniadakis, foundational PINN framework: https://www.sciencedirect.com/science/article/pii/S0021999118307125. Authors' equations and examples: https://maziarraissi.github.io/PINNs/. Krishnapriyan et al., Characterizing possible failure modes in physics-informed neural networks: https://proceedings.neurips.cc/paper/2021/hash/df438e5206f31600e6ae4af72f2725f1-Abstract.html. The decay calculation and enclosure case are original illustrations.

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