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SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks
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Proposes SNN-PDE, a graph/simplicial neural network that combines Hodge-Laplacian simplicial convolution with multi-hop graph convolutions to learn time-dependent PDE dynamics on irregular grids.
Problem
Learning surrogate solvers for time-dependent PDEs on irregularly spaced, unstructured measurement locations and manifold-like domains, where grid-based and pairwise GNN methods miss higher-order structure.
Approach
Introduces discrete Hodge-Laplacian simplicial convolutions (over edges and triangles, fused with attention) alongside dual multi-hop grid-structural and adaptive graph convolutions, and uses the resulting node embeddings as the right-hand side of a neural ODE solved with an adaptive Adams solver.
Key results
- SNN-PDE achieves the lowest mean relative error on all five dynamical-system datasets in Table 1, e.g., 0.108 vs 0.150 (R-MPGNN) on ConvDiff N750 and 0.022 on Burgers.
- On the Eastern U.S. wildfire dataset SNN-PDE reports 64.424 vs 71.656 for the best runner-up, and on Western U.S. 200.657 vs 214.041.
- Ablations show removing the multi-channel simplicial encoder raises ConvDiff N750 error from 0.108 to 0.135 and Western U.S. error from 200.657 to 227.515.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use SNN-PDE instead of the strongest baseline | PDE state estimation · ConvDiff N750 | mean relative error: 0.119±0.015 -> 0.108±0.010 | Helped |
| use SNN-PDE instead of the strongest baseline | PDE state estimation · ConvDiff N1500 | mean relative error: 0.078±0.012 -> 0.072±0.004 | Helped |
| use SNN-PDE instead of the strongest baseline | PDE state estimation · ConvDiff N3000 | mean relative error: 0.079±0.036 -> 0.049±0.006 | Helped |
| use SNN-PDE instead of the strongest baseline | PDE state estimation · Heat | mean relative error: 0.052±0.036 -> 0.022±0.006 | Helped |
| use SNN-PDE instead of the strongest baseline | PDE state estimation · Burgers | mean relative error: 0.330±0.010 -> 0.306±0.050 | Helped |
| use SNN-PDE instead of the strongest baseline | wildfire PDE state estimation · Eastern U.S. | mean relative error: 71.656±4.164 -> 64.424±2.361 | Helped |
| use SNN-PDE instead of the strongest baseline | wildfire PDE state estimation · Western U.S. | mean relative error: 214.041±0.980 -> 200.657±0.907 | Helped |
| use SNN-PDE instead of the strongest baseline | PDE state estimation · Stanford Bunny | mean relative error: 0.217±0.005 -> 0.213±0.001 | Helped |
| add the multi-channel simplicial convolution encoder (ablation) | SNN-PDE · ConvDiff N750 | mean relative error: 0.135±0.037 -> 0.108±0.010 | Helped |
| add the multi-hop grid-structural graph convolution (ablation) | SNN-PDE · ConvDiff N750 | mean relative error: 0.113±0.025 -> 0.108±0.010 | Helped |
| add the multi-hop adaptive graph convolution (ablation) | SNN-PDE · ConvDiff N750 | mean relative error: 0.116±0.023 -> 0.108±0.010 | Helped |
| add the multi-channel simplicial convolution encoder (ablation) | SNN-PDE · Burgers | mean relative error: 0.336±0.032 -> 0.306±0.050 | Helped |
Surprising findings
- GNN-PDE has a lower mean relative error than SNN-PDE on the Stanford Bunny dataset, though with much larger variance.
Limitations
- Method assumes PDEs on R^d / graphs; a formal treatment of PDEs on general manifolds is left to future work (Remark 1).
- On Stanford Bunny, GNN-PDE has a lower mean error than SNN-PDE.
- Results are reported as mean over 5 runs with limited hyperparameter search ranges; wildfire evaluation uses only 10 epochs.
Experimental setup
Datasets: ConvDiff synthetic simulations Heat equation simulations Burgers equation simulations HYSPLIT PM2.5 simulations (Quebec wildfire, NOAA) Stanford Bunny
Benchmarks: ConvDiff N750 ConvDiff N1500 ConvDiff N3000 Heat Burgers Eastern U.S. wildfire Western U.S. wildfire Stanford Bunny
Metrics: mean relative error standard deviation one-sided two-sample t-test
Baselines: CNN ResNet EdgeConv PDE-Net 2.0 GNN-PDE R-MPGNN EA-GNN M-GNN
Builds on: MP-PDE GNN-PDE simplicial neural networks EdgeConv PDE-Net 2.0
Closest papers by meaning
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.89)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.87)
- P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics (NeurIPS 2024 · similarity 0.87)
- Mechanistic PDE Networks for Discovery of Governing Equations (ICML 2025 · similarity 0.86)
- Mesh-Free Convolution: Learned Spectral Attenuation and Transport (NeurIPS 2026 · similarity 0.86)
- Graph Neural PDE Solvers with Conservation and Similarity-Equivariance (ICML 2024 · similarity 0.85)
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