DeepONet + 1D Poisson + Laplace¶

Notebook: DeepONet_Poisson1D_Laplace_Tutorial.ipynb

This tutorial builds a 1D scientific machine learning example around the boundary value problem

It uses a DeepONet1D surrogate to learn the field-to-field map from a forcing field \(f(x)\) to the equilibrium response field \(u(x)\), then adds a last-layer Laplace approximation to quantify predictive uncertainty.

What the notebook covers:

• physical meaning of the Poisson problem in diffusion / elasticity / electrostatics language,
• dataset generation from random smooth forcing fields,
• DeepONet branch/trunk structure for 1D operator learning,
• MAP training diagnostics,
• uncertainty bands over the 1D spatial coordinate,
• in-domain vs OOD comparisons for uncertainty calibration.

Why this notebook is useful:

• both the input and the output are 1D fields, so the neural-operator map is easy to interpret,
• uncertainty is visible as a standard mean-curve plus confidence-band plot,
• the Poisson solver is stable and inexpensive, which keeps the tutorial focused on operator learning rather than PDE debugging.