Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation
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Coarsens unstructured 2D grids by k-means pooling followed by gradient-based optimization of coarse site locations through a differentiable Voronoi tessellation and graph-based finite volume solver, matching fine-grid pressure responses for slightly compressible porous flow and the wave equation.
Problem
Reducing the size of discrete unstructured-grid PDE simulations (e.g., subsurface flow) while preserving the accuracy of the modelled variables at points of interest.
Approach
Combines k-means clustering pooling with end-to-end differentiable Voronoi tessellation (circumcenter/shoelace geometry via autodiff) and a differentiable graph FV solver, so coarse site positions are optimized to minimize pressure-time-series RMSE against the fine-grid simulation, without training data.
Key results
- On the 90000-to-1000 point coarsening, the optimized grid reaches RMSE about 0.042 versus 0.057 for k-means + average pooling after 60 epochs, with simulation 34x faster.
- The method reduces grid points up to 10x on the loop scenario while preserving pressure dynamics at the sink point.
- Scalability: a 1,000,000-point simulation takes 239 minutes with 4.9 GB peak RAM on two CPU cores in Google Colab.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Optimize the coarsened grid with differentiable simulation instead of using k-means + average pooling alone | 90000→1000 coarsening · 90000→1000 coarsening | RMSE: 0.057 -> approximately 0.042 | Helped |
| Coarsen the point cloud from 90,000 to 1,000 points | Simulation · 90000→1000 coarsening | speedup: 34x | Helped |
Limitations
- Demonstrated only on 2D problems with explicit forward Euler time stepping, which has stability restrictions.
- Optimization requires many epochs and gradient computation through long simulation graphs, with cost growing with grid size and time steps.
- Evaluation is on synthetic and limited real-world examples with few measurement points and no broad benchmark comparison.
Experimental setup
Datasets: synthetic loop scenario point cloud (312 points) synthetic 90000-point permeability field real-world oil reservoir permeability field (petroleum example)
Metrics: RMSE MSE runtime (minutes) peak RAM (GB)
Baselines: k-means + average pooling
Builds on: PyTorch Geometric PyTorch k-means
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