Navier–Stokes (FEniCSx) example

This is an advanced example showing how to apply the gp_active_mcmc workflow to a PDE-based forward model (incompressible Navier–Stokes) on a backward-facing-step geometry.

It is not executed as part of the documentation build because it requires a separate FEniCSx/DOLFINx + PETSc + MPI + Gmsh environment.

Requirements

The Navier–Stokes example uses the FEniCS/DOLFINx ecosystem and a standard MPI toolchain. These are typically installed via conda-forge (not reliably via pip):

fenics-dolfinx=0.9.0 (conda-forge, version=0.10 does not work)
mpich (conda-forge)
gmsh=4.15.0
python-gmsh=4.15.0
pyvista

A representative conda installation (adjust to your environment):

conda install -c conda-forge fenics-dolfinx=0.9.0 mpich gmsh=4.15.0 python-gmsh=4.15.0 pyvista

Where to find the code

The full example lives in the repository under:

examples/navier-stokes/utils/ — mesh, solver, QoI extraction, animation utilities
examples/navier-stokes/run_forward_*.py — build/validate a POD–GP surrogate
examples/navier-stokes/run_backward_*.py — inverse problem with Active/Adaptive MCMC

Start here:

Forward (surrogate): examples/navier-stokes/run_forward_ns_hf_surrogate.py
Backward (inference): examples/navier-stokes/run_backward_ns_active_mcmc.py
MF/HF solver sweep: examples/navier-stokes/cfd/run_mf.py

Precomputed outputs (from the example)

The figures below are generated by running the scripts in examples/navier-stokes/ and are committed as static assets so they render on the hosted documentation.

Velocity magnitude animation

Outlet profiles (QoI) for representative parameters

Outlet profiles for various upstream height Outlet profiles for various inlet velocities

POD energy (surrogate compression)

POD energy

Surrogate prediction examples (median and worst)

Surrogate examples median Surrogate examples worse

Noisy Observation

Noisy observation

Posterior using MCMC Active Learning

Not reliable and seed-dependent.

Samples using MCMC-Active Learning HF Fraction using MCMC-Active Learning Prediction after MCMC sampling

Posterior using DA-MCMC Active Learning with Adaptive Subchain

Robust and accurate.

Samples using DA-MCMC-Active Learning HF Fraction using DA-MCMC-Active Learning Subchain Lengths using DA-MCMC-Active Learning Prediction after MCMC sampling

Summary

Active MCMC summary: {'n_steps': 901, 'n_dim': 3, 'move_fraction': 0.04666666666666667, 'hf_call_fraction': 0.003329633740288568, 'n_hf_calls': 3, 'posterior_rmse': 0.02565835473083319}

Adaptive DA-MCMC summary: {'n_steps': 900, 'n_dim': 3, 'move_fraction': 0.23804226918798665, 'hf_call_fraction': 0.025555555555555557, 'n_hf_calls': 23, 'mean_subchain_length': 129.20529801324503, 'posterior_rmse': 0.011547864268091526}