Choose the physical initial-boundary-value problem first; decide how to enforce it in the PINN second. Hard constraints can make representable conditions hold by construction, while soft constraints are easier to adapt to noisy or complex data but only penalize violations. Neither method can rescue boundary or initial data that do not describe the intended flow, and the available studies do not establish a universal winner.
Specify the flow problem before designing the PINN
Start by defining the domain, whether the target is steady or transient, the fluid assumptions, the variables to be predicted, and the measurements or other evidence available for validation. Then divide the domain boundary into segments and identify each segment’s physical role. Conditions should represent the experiment or model—not be chosen merely because they are convenient to encode.
- Walls: Identify which are stationary and which move. Prescribe the velocity behavior appropriate to the wall model.
- Inlets: State the incoming flow data, such as a specified velocity profile or flow rate, as appropriate to the problem.
- Outlets: Choose an outlet condition consistent with the intended physical or computational setup. Do not assume that every formulation requires the same pressure condition.
- Symmetry boundaries: Specify the symmetry relations appropriate to the variables and geometry.
- Periodic boundaries: Pair the corresponding boundary segments and prescribe the required matching across them.
These are roles, not a universal recipe: the actual condition depends on the flow, geometry, and formulation. Also decide how pressure is referenced in the chosen model. In the velocity-pressure formulation described by NSFnets, pressure is a hidden state inferred through incompressibility; that approach does not require a separate pressure boundary or initial condition. This should not be generalized to every Navier–Stokes formulation.
Decide whether the problem needs initial data
Transient flow
A time-dependent problem needs an initial velocity field over the spatial domain at the chosen initial time. Check that the field is physically plausible, compatible with incompressibility and imposed fluxes, and consistent with boundary values where the initial surface meets the boundary. Compatibility can depend on the geometry and flow class; there is no single checklist that guarantees it for every problem.
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Steady flow
A steady formulation has no temporal initial condition. Do not add one simply because a PINN implementation supports an initial-data loss. Prescribe the boundary data for the steady problem and enforce the governing equations for that regime.
Choose how the PINN enforces the conditions
| Approach | How it works | Best reason to consider it | Main caution |
|---|---|---|---|
| Soft | Add boundary and, for transient problems, initial-condition residuals to the loss at sampled points. | Values are noisy or uncertain, or the geometry and data are awkward to encode exactly. | Penalties only encourage satisfaction. Results depend on point sampling and on the relative scales and weights of the losses. |
| Hard | Construct the trial function or network output so selected conditions hold by construction. | Conditions are known and can be represented cleanly, and exact satisfaction is important. | An unsuitable ansatz can exclude valid solutions or be impractical for complex geometry, corners, mixed conditions, or changing data. Its derivatives must also work with the PDE residual. |
| Hybrid | Combine a preliminary soft solution with a more boundary-aware mechanism during refinement. | A simple hard representation is awkward, but boundary violations still need focused treatment. | It is another option to test, not a demonstrated general improvement over a well-tuned soft baseline. |
Soft enforcement: flexible penalties
For soft enforcement, sample points on each relevant boundary segment and, for a transient problem, on the initial-time surface. Add the corresponding residuals to the objective alongside the governing-equation residual. This approach accommodates uncertain or noisy values, but reducing a penalty does not guarantee exact satisfaction. In some settings, boundary penalties can be less robust or fail to converge to the desired solution; treat that as a risk to evaluate, not an inevitable outcome.
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Hard enforcement: build conditions into the trial function
For a simple homogeneous Dirichlet condition, a factor that vanishes on the constrained boundary can multiply a free neural-network output. For nonhomogeneous data, a lifting term can supply the prescribed boundary value, while a boundary-vanishing factor gates the unconstrained part. In schematic form, this is a prescribed lifting plus a boundary-vanishing factor times a network output.
Hard enforcement is useful only when the representation fits the actual conditions. Check that it does not rule out valid solutions, that it handles every constrained segment, and that it remains sufficiently smooth: Navier–Stokes residuals involve spatial derivatives. A construction that is easy on a simple domain may become difficult with corners, mixed boundary types, or complicated nonhomogeneous data. Published demonstrations of hard constraints for selected steady-flow and complex-boundary cases establish feasibility in those setups, not universal superiority.
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Hybrid enforcement: a problem-specific alternative
A reported hybrid approach uses soft training to obtain a preliminary solution, then a stronger boundary-aware mechanism for refinement. It has been studied for a cylinder wake and a blocked cavity with a segmented inlet. Those examples make hybrid enforcement worth testing when direct hard encoding is awkward; they do not show that it will outperform a carefully tuned soft approach on a different problem.
Make the choice against the actual problem
Compare candidate approaches on the conditions and outputs that matter, rather than on a single aggregate training loss.
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- Condition exactness: Do the required initial and boundary values need to hold by construction, or is a measured, approximate fit acceptable?
- Data quality: Are values trusted and precise, or noisy, uncertain, or incomplete?
- Representability: Can the geometry and nonhomogeneous values be encoded without excluding valid solutions?
- Differentiability: Does the construction remain smooth enough for the spatial derivatives used in the PDE residual?
- Optimization behavior: How sensitive is soft enforcement to loss weights and sampling? Does a hard construction make the trial function impractical?
- Flow accuracy: Does the method reproduce the quantities of interest, not just reduce its training objective?
Where possible, compare predictions with trusted CFD, analytical solutions, or experimental measurements. Report initial-condition and boundary-condition errors separately from interior PDE and incompressibility residuals. Inspect walls, corners, and other high-gradient regions, as well as relevant profiles, pressure, forces, or other target outputs. A low total training loss alone is not proof that the flow is correct.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Interpret reported accuracy in context
A 2025 preprint by Ritik Pal, Soubhik Mukherjee, Urmi Dutta, and Arghya Choudhury reports normalized L2 errors in the range O(10-4)–O(10-1) for its chosen case studies. That range is specific to those cases; it is not a typical-accuracy promise or a guarantee for another geometry, Reynolds number, formulation, or data quality. The available studies are case-specific rather than a controlled comparison across all such settings, so they do not support a claim that hard, soft, or hybrid enforcement is always faster or more accurate.
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