👉 Existing students can access this tutorial anytime from their course dashboard.

We just added a fresh, implementation-first tutorial to my PINNs course — and this one is special.

In this tutorial, we solve an inverse problem for the 2D incompressible Navier–Stokes equations using a real CFD dataset (cylinder wake flow). We intentionally keep only sparse velocity observations and ask a PINN to reconstruct the full flow physics:

• learn the fields (u, v, p)
• and identify unknown physical coefficients (C1, C2) directly from data + equations

If you’ve ever wondered whether PINNs can actually infer physics hidden inside CFD data — this notebook is built to verify that, end-to-end.

What you’ll learn (the practical stuff)

This tutorial is structured like a real engineering workflow, not a demo:

1) Dataset → training pool (properly)

• Load the .mat cylinder wake dataset and validate shapes
• Select target time snapshots cleanly
• Plot reference u, v, p on an unstructured grid (so you can see the “truth” before training)

2) Collocation points: where physics is enforced

• Construct physics collocation points in space-time
• Understand how the PINN “sees” the PDE without being handed full fields everywhere

3) The inverse PINN model (PyTorch)

• Build an MLP that outputs (u, v, p)

• Define Navier–Stokes residuals using autodiff:

  • continuity + x-momentum + y-momentum
• Make C1 and C2 trainable parameters and learn them during training

4) Training + validation

• Two-stage Adam training (practical, stable)
• Track separate data loss vs physics loss
• Compare the learned solution against the reference CFD fields

A key concept covered clearly: pressure gauge freedom

The notebook also explains why pressure is only identifiable up to an additive constant (i.e., (p + C) is equally valid), and what that means when comparing PINN pressure with CFD pressure.

If you’re already in the course, you can jump into the notebook now and run it as-is. If you’re serious about PINNs, this tutorial will feel like a proper bridge into research-grade problems.

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For readers who want to understand how PINNs are formulated in practice for the incompressible Navier–Stokes equations — including learning unknown flow parameters, coupling velocity and pressure losses, scaling momentum vs continuity residuals, and why incorrect physical coefficients can still produce low Navier–Stokes residuals — a hands-on walkthrough is available as part of the PINNs Masterclass.

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