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Choose a characteristic length L and velocity U, then scale position by L, time by L/U, velocity by U, and pressure by ρU². For constant-density, incompressible Newtonian flow with constant kinematic viscosity ν and no separately retained body force, this gives a dimensionless momentum equation with viscous coefficient 1/Re, where Re = UL/ν. A PINN uses that equation to form residual losses alongside the initial, boundary, and observational constraints for the particular problem.
Choose characteristic scales that match the flow
Let L be a representative length from the geometry or flow, and let U be a representative velocity. These are modeling choices, not universal constants: state what they mean for your problem and use them consistently in the equations, data, and conditions.
Define dimensionless variables with a star:
- x = Lx*, so x* = x/L.
- t = (L/U)t*, so t* = tU/L.
- u = Uu*, so u* = u/U.
- p = ρU²p*, so p* = p/(ρU²).
Here ρ is the constant fluid density. The time scale L/U is the time required to travel one characteristic length at speed U.
Derive the dimensionless incompressible equations
For a constant-density, incompressible Newtonian fluid with constant kinematic viscosity ν, and with no separately retained body-force term, start from
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∂u/∂t + (u·∇)u = −(1/ρ)∇p + ν∇²u, ∇·u = 0.
Under the scales above, each inertial term has scale U²/L. The viscous term has scale νU/L². Dividing the momentum equation by U²/L gives
∂u*/∂t* + (u*·∇*)u* = −∇*p* + (1/Re)∇*²u*, ∇*·u* = 0,
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where ∇* differentiates with respect to dimensionless position and
Re = UL/ν.
The Reynolds number compares inertial and viscous effects under the chosen scales; its reciprocal is the coefficient on the dimensionless viscous term. The velocity-pressure form with this 1/Re coefficient is used in the NSFnets treatment of incompressible Navier–Stokes equations (NSFnets (2020)).
Do not silently drop other physics
If the model includes body forces, variable material properties, compressibility, or additional physical effects, retain those terms and scale them too. The displayed equation is not the right complete model when those effects matter. Pressure can also be scaled with a different physically justified pressure scale; if you do, show the resulting coefficient on the pressure-gradient term rather than assuming it remains one.
Transform the conditions and data as well
Nondimensionalization applies to the full PINN problem, not just the interior PDE. Convert the initial and boundary conditions using the same reference scales. For example, a dimensional velocity condition u = ub becomes u* = ub/U; a dimensional pressure condition p = pb becomes p* = pb/(ρU²). Convert coordinates and times to x* and t* too. Observational data should likewise be represented in the units and variables predicted by the network.
A cylinder example in the foundational PINN paper specifies “a non-dimensional free stream velocity u∞ = 1, cylinder diameter D = 1, and kinematic viscosity ν = 0.01.” Those values describe that paper’s setup, not default scales or parameters for other flows (Raissi, Perdikaris, and Karniadakis (2019)).
Build the PINN residuals from dimensionless fields
For a velocity-pressure network that predicts, for example, u*, v*, and p*, use automatic differentiation with respect to x*, y*, and t* to form the dimensionless continuity and momentum residuals at interior collocation points. A representative objective is
Ltotal = λmomLmom + λcontLcont + λICLIC + λBCLBC + λdataLdata.
The terms represent momentum residual, continuity residual, initial-condition, boundary-condition, and observation losses. Include only the terms relevant to the problem. One Navier–Stokes PINN study, for example, itemizes velocity, PDE-residual, nodal reference-pressure, and boundary-condition losses (Active training of physics-informed neural networks (2021)). The original PINN paper describes mean-squared-error minimization and automatic differentiation through TensorFlow’s gradient graph, but the formulation does not require that specific software framework (Raissi, Perdikaris, and Karniadakis (2019)).
Make the loss definition inspectable
For each loss component, record how its residuals are reduced and weighted:
- Specify whether the mean squared error is averaged over collocation points, data points, or both.
- State whether residual components are divided by a characteristic residual scale before squaring.
- Show whether momentum components receive separate weights or are combined into one term.
- Monitor the component losses and, where possible, their influence on parameter updates during training.
Nondimensionalizing the equations does not automatically make every loss contribute appropriately during optimization. NSFnets examines weighting between data and physics terms and describes dynamic weighting; the evidence does not establish one universally best weighting or a general numeric training improvement caused by nondimensionalization alone (NSFnets (2020)).
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Choose a formulation for the outputs and available constraints
In addition to velocity-pressure (VP), NSFnets presents a vorticity-velocity (VV) formulation for unsteady incompressible flow. Neither is universally superior; choose based on what the network must predict, what data and boundary conditions you have, and what derivatives the residual requires (NSFnets (2020)).
| Decision point | Velocity-pressure (VP) | Vorticity-velocity (VV) |
|---|---|---|
| Network outputs | Velocity and pressure are predicted fields. | Velocity and vorticity are predicted fields; pressure is not a predicted output in this form. |
| Residual derivatives | Includes pressure-gradient terms and velocity derivatives in momentum, plus velocity divergence for continuity. | Uses the vorticity-velocity equations; the required derivatives differ from VP and should be assessed for the specific implementation. |
| Data and conditions | Fits naturally when velocity and pressure observations or constraints are available and pressure is a required result. | Fits when available measurements and conditions support velocity and vorticity, or when that representation suits the problem. |
| Choice to check | Account for pressure data or reference constraints and enforce incompressibility through continuity. | Check that vorticity-related information and boundary conditions are represented adequately for the chosen equations. |
The table describes formulation-level distinctions; the appropriate derivative implementation and boundary treatment depend on the specific problem and formulation.
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