FNO2D + Darcy Flow + Laplace¶

Notebook: FNO2D_Darcy_Laplace_Tutorial.ipynb

This tutorial builds a 2D scientific machine learning example around the Darcy-flow equation

with Dirichlet boundary condition

The notebook uses three input fields and one output field:

• permeability k(x,y),
• source / sink field q(x,y),
• boundary-condition field g(x,y),
• pressure / hydraulic head u(x,y).

What the notebook covers:

• physical interpretation of Darcy flow and the role of each input field,
• notebook-local dataset generation with a variable-coefficient finite-difference solver,
• a reusable FNO2D model from deepuq.models,
• residual training around a deterministic Darcy baseline,
• last-layer Laplace approximation with block_diag,
• optional comparison against lowrank_diag,
• uncertainty maps comparing in-domain and OOD permeability / forcing regimes.

Why this tutorial matters:

• it is the package's first 2D Fourier-neural-operator example,
• it uses a realistic elliptic PDE rather than only time-evolution examples,
• it shows how uncertainty increases when permeability contrast and forcing patterns move outside the training distribution.

Primary references cited in the notebook:

• Zongyi Li et al., Fourier Neural Operator for Parametric Partial Differential Equations.
• Lu Lu et al., Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.
• David J. C. MacKay, A Practical Bayesian Framework for Backpropagation Networks.
• Hippolyt Ritter et al., A Scalable Laplace Approximation for Neural Networks.
• Erik Daxberger et al., Laplace Redux - Effortless Bayesian Deep Learning.