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FNO3D + 3D Heat Equation + Laplace

Notebook: FNO3D_Heat_Laplace_Tutorial.ipynb

This tutorial builds a 3D scientific machine learning example around the periodic heat equation

\[ \partial_t u = \kappa \Delta u, \qquad (x,y,z)\in[0,1)^3, \]

and learns the operator

\[ \mathcal{G}: u_0(x,y,z) \mapsto u(x,y,z,T) \]

with a reusable FNO3D model from deepuq.models.

What the notebook covers:

  • physical meaning of the 3D heat-diffusion problem,
  • exact spectral data generation for train/test/OOD splits,
  • a 3D Fourier Neural Operator with spectral convolution layers,
  • a residual physics baseline that keeps the CPU tutorial accurate,
  • last-layer Laplace approximation with block_diag,
  • predictive mean and epistemic-uncertainty slice plots across the 3D grid.

Why this notebook is useful:

  • it is the package's first 3D operator-learning example,
  • the PDE has an exact spectral solution, so the tutorial can focus on learning and UQ rather than numerical solver noise,
  • uncertainty can be inspected on central xy, xz, and yz slices and compared between in-domain and OOD inputs.

Executed quick-mode result in this repository:

  • MAP test RMSE: about 1.0e-5
  • Laplace test RMSE: about 7.9e-4