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Graph Operator + Gray-Scott + Deep Ensembles

Notebook: GraphOperator_GrayScott_Ensemble_Tutorial.ipynb

This tutorial introduces a graph-based neural operator for scientific machine learning using a Gray-Scott reaction-diffusion system. The main model is GraphNeuralOperator2D, which treats a regular Cartesian grid as a graph and applies local message passing over node embeddings.

Primary task:

  • input: two-species state [A(x,y,t), B(x,y,t)]
  • output: one-step-ahead state [A(x,y,t+\Delta t), B(x,y,t+\Delta t)]

Primary UQ method:

  • DeepEnsembleWrapper over multiple independently trained graph operators

Optional comparison:

  • last-layer Laplace with LaplaceWrapper(..., subset_of_weights="last_layer", hessian_structure="block_diag")

Why this tutorial matters:

  • it is the package's first graph-based neural-operator example,
  • it shows how to use a graph model even when the underlying data lives on a regular grid,
  • it compares in-domain and held-out Gray-Scott regimes,
  • it visualizes field-wise predictive mean and epistemic standard deviation for both species.

Dataset note:

  • if a local The Well Gray-Scott archive is available, the notebook loads it through deepuq.data.the_well,
  • otherwise quick mode falls back to a small synthetic Gray-Scott generator so the notebook remains executable in a self-contained way.

Primary references cited in the notebook:

  • Li et al., Fourier Neural Operator for Parametric Partial Differential Equations
  • Battaglia et al., Relational inductive biases, deep learning, and graph networks
  • Pfaff et al., Learning Mesh-Based Simulation with Graph Networks
  • Lakshminarayanan et al., Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles
  • MacKay (1992), Ritter et al. (2018), and Daxberger et al. (2021) for the optional Laplace comparison