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1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problemsSet up the inverse problem as a joint fit: train a neural network to match measured velocities while also minimizing the Navier–Stokes momentum residuals, and optimize the unknown equation coefficients alongside the network weights. For a reproducible starting point, use the published two-dimensional cylinder-wake case at Reynolds number 100—but treat its data volume and training settings as one benchmark, not a guarantee for other flows.
What does the inverse PINN estimate?
In a forward flow problem, the governing equations and their parameters are specified and the goal is to predict the flow. In an inverse problem, some information is missing: velocity observations are available, but one or more coefficients in the equations must be inferred. A physics-informed neural network (PINN) represents the flow with a differentiable neural network, then trains that representation against both the observations and the governing equations.
For incompressible two-dimensional flow, let u(t,x,y) and v(t,x,y) be the velocity components and p(t,x,y) the pressure. A common inverse formulation treats two momentum-equation coefficients, λ1 and λ2, as unknowns. In a nondimensional form, the momentum residuals can be written as:
f = ut + λ1(u ux + v uy) + px − λ2(uxx + uyy)
g = vt + λ1(u vx + v vy) + py − λ2(vxx + vyy)
The target is for both residuals to be near zero at the points where the equations are enforced. Depending on the equation scaling and nondimensionalization, the learned coefficients map to physical quantities differently; define that mapping before interpreting a fitted coefficient as viscosity. The model must also satisfy incompressibility, ux + vy = 0.
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How should incompressibility be enforced?
One option is to predict a stream function ψ and pressure p, then derive velocity as u = ψy and v = −ψx. With the smoothness needed for these derivatives, the divergence-free condition follows from equality of mixed derivatives, rather than from a separate approximate penalty. This is the formulation used in the 2019 cylinder-wake inverse study by Raissi, Perdikaris, and Karniadakis.
A second option is to predict u, v, and p directly and add a continuity residual to the loss. These choices trade implementation simplicity against how continuity is handled:
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| Representation | Continuity | Derivative and implementation implications |
|---|---|---|
| Stream function and pressure | Enforced by construction through u = ψy, v = −ψx. | Velocity is itself differentiated to form momentum residuals, so higher-order derivatives of the network output are needed. The velocity observations must be compared with these derived quantities. |
| Direct velocity components and pressure | Usually encouraged through an explicit continuity residual, unless another constraint is used. | Momentum residuals use derivatives of directly predicted velocities; continuity adds its own derivatives and loss term. This can be simpler to express, but divergence-free behavior is not automatic. |
Neither parameterization is universally better. Choose according to the geometry, available boundary information, differentiability and computational cost, and how strongly the application requires exact continuity.
How do you build and train the inverse problem?
- Define the physical model and unknowns. Write the momentum equations, continuity constraint, domain, initial and boundary conditions, and the coefficients to estimate. State whether variables are dimensional or nondimensional. In the published benchmark, free-stream speed and cylinder diameter are each set to 1, and kinematic viscosity is 0.01, giving Reynolds number 100.
- Choose network outputs. For the stream-function formulation, the network maps (x,y,t) to (ψ,p); derive u and v from spatial derivatives. For a direct-velocity formulation, the network maps coordinates to (u,v,p). The benchmark uses a three-output network in its DeepXDE example; its documentation and script should be checked for the exact formulation and version being reproduced.
- Differentiate to form the residuals. Use automatic differentiation to calculate time and spatial derivatives, including second spatial derivatives in the momentum equations. Assemble each momentum equation with all terms on one side, producing residuals such as f and g. Automatic differentiation is central to the original PINN framework; finite differences are not needed to define these network derivatives.
- Define separate data and physics losses. A basic objective is L = wdataLdata + wphysicsLphysics, where the data term measures mismatch between predicted and observed velocities, and the physics term penalizes nonzero momentum residuals at equation-enforcement points. Include continuity as a residual term if it is not built into the representation. Apply boundary or initial-condition losses when the problem requires them. The relative weights, normalization, and units need to be chosen for the specific problem; the cited sources do not prescribe a universal weighting rule.
- Optimize coefficients and network parameters together. Treat the unknown coefficients as trainable variables in the same optimization as the neural-network weights and biases. The optimizer then seeks a flow representation that explains the observations while making the governing-equation residuals small. A low training loss alone does not establish that the inferred coefficients are identifiable or correct.
- Validate against withheld information. Reserve velocity observations from training and assess their prediction error. Inspect residual behavior separately, and compare recovered coefficients with known values in a benchmark or independently justified values in an application. Also examine sensitivity to sampling, loss scales, and noise; a model can fit observations without uniquely identifying every unknown.
What does the Re=100 cylinder-wake benchmark establish?
Raissi, Perdikaris, and Karniadakis reported an inverse cylinder-wake demonstration at Reynolds number 100. They randomly selected 5,000 velocity observations, which they described as 1% of their high-resolution dataset, and retained the remaining observations for validation. Those figures describe that study’s dataset and split; 5,000 is not a general minimum sample size.
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The benchmark demonstrates the joint-inference setup: velocity data constrain the learned flow, while the momentum equations constrain its dynamics and permit coefficient estimation. It does not establish expected accuracy for a different geometry, observation-noise level, boundary condition, or flow regime. In particular, equation residuals do not by themselves prove that the parameters can be uniquely recovered from a chosen set of measurements.
How can you reproduce the setup with DeepXDE?
DeepXDE is one practical library route: its documentation covers PINNs for forward and inverse differential-equation problems, and its repository includes an inverse Navier–Stokes cylinder example. The example is a baseline to inspect and reproduce, not a claim that its settings are optimal.
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| Example setting | Value in the cited repository script |
|---|---|
| Flow case | Two-dimensional cylinder flow at Re=100. |
| Inputs and network | Inputs (x,y,t); fully connected network with 6 hidden layers and 50 units in each hidden layer. |
| Activation and initialization | Tanh activation; Glorot uniform initialization. |
| Network outputs | Three outputs. |
| Configured point counts | 700 domain points, 200 boundary points, and 100 initial points. |
| Training stages | Two Adam stages: learning rate 1e-3 for 10,000 iterations, then 1e-4 for 10,000 iterations. |
| Backend options listed by DeepXDE | TensorFlow, TensorFlow compat v1, PyTorch, and Paddle. |
The repository script also loads measured velocity data, defines a space-time domain, constrains observed velocity components, and makes two PDE coefficients trainable. Before reproducing it, select and record the DeepXDE version or branch and confirm backend compatibility: versioned documentation and code can change. Confirm that the example’s output parameterization and loss construction match the equations and unknowns in your own problem before adapting its point counts or iteration schedule.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How should pressure and viscosity estimates be interpreted?
Pressure has a reference ambiguity
In the cited inverse cylinder example, pressure is recovered only up to an additive constant. Pressure gradients enter the momentum equations, so shifting pressure by a constant does not change those gradients. To report absolute pressure, choose and state a reference convention, such as fixing pressure at a point or specifying a reference level. Without that convention, compare pressure differences or gradients rather than treating the offset as physically identified.
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Viscosity depends on the equation scaling
In the benchmark’s stated nondimensional setting, free-stream speed and diameter are 1 and kinematic viscosity is 0.01. When estimating viscosity in another setup, verify how the coefficient multiplies the diffusion term in the precise equations used by the code, and convert between that coefficient and dimensional viscosity using the chosen scales. A fitted coefficient is not automatically a dimensional material property.
What should you check when training is unreliable?
- Parameter identifiability: determine whether the observations and constraints contain enough information to distinguish the unknown coefficients. A small residual is not proof of a unique solution.
- Boundary and initial conditions: ensure that the model represents the actual physical constraints and that their losses or exact constraints are applied consistently.
- Units and nondimensionalization: scale coordinates, time, velocities, pressure, and equation terms coherently. Inconsistent scales can make nominally comparable loss terms behave very differently.
- Loss balance: inspect data and physics losses separately. Choose weights and normalization for the problem rather than assuming one fixed ratio works for all flows.
- Observation quality and sampling: keep validation data separate and test whether estimates change with sampling or plausible noise. The published 1% split is a benchmark design, not a prescription.
- Pressure reference: impose a gauge convention if absolute pressure is required; otherwise assess identifiable quantities such as pressure gradients.
- Reproducibility: record the framework version or repository branch, backend, architecture, sample construction, initialization, loss definitions, and optimizer schedule.
PINNs are used across fluid mechanics, but the literature also examines training pathologies, sampling strategies, domain decomposition, and uncertainty quantification. Those are possible avenues for a specific difficult case, not automatic fixes or evidence that a particular architecture or optimizer will work for every application.
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