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Graph Neural PDE Solvers with Conservation and Similarity-Equivariance
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Introduces FluxGNN, a graph neural PDE solver built on finite volume flux functions that enforces local conservation and similarity-equivariance, and shows better spatial out-of-domain generalization than MP-PDE and PENN.
Problem
Data-driven PDE solvers often violate conservation laws and symmetries and fail to generalize to unseen spatial domains, shapes, and scalings.
Approach
Shows mathematically when message-passing GNNs are conservative (antisymmetric messages, identity update, linear encoder/pseudoinverse decoder) and builds FluxGNN as a learned numerical-flux finite volume scheme with permutation-invariant flux functions and similarity-equivariant MLPs using local scales (Courant-number-like inputs).
Key results
- On convection-diffusion, FluxGNN reaches MSE 3.08e-5 vs 1408.47e-5 for FVM on the test set, with conservation error on the order of 1e-8 comparable to FVM.
- On Navier-Stokes with mixture, FluxGNN keeps nearly identical loss on the taller dataset (u loss 1.184e-1) while MP-PDE degrades to 8.377e-1 for u; PENN has 1.476e-1 for u on taller.
- On the scaling dataset PENN diverges (NaN) while FluxGNN gives u loss 1.219e-1, almost unchanged from test.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace identity trainable functions in the FVM solver with learned FluxGNN functions | Convection–diffusion solver · Convection–diffusion test | MSE loss: 1408.47 ± 32.75 -> 3.08 ± 0.08 | Helped |
| Use FluxGNN instead of FVM | Convection–diffusion solver · Convection–diffusion test | Conservation error: 1.63 ± 0.19 -> 1.75 ± 0.32 | No clear effect |
| Use FluxGNN instead of FVM, MP-PDE, or PENN | Navier–Stokes mixture solver · Navier–Stokes mixture test | Conservation error α: 0.06 ± 0.03 vs 0.01 ± 0.00; 1001.95 ± 120.43; 1356.62 ± 283.00 | Helped |
| Use FluxGNN instead of PENN | Navier–Stokes mixture solver · Navier–Stokes mixture test | Loss α: 0.358 ± 0.005 -> 0.349 ± 0.005 | Helped |
| Use FluxGNN instead of PENN | Navier–Stokes mixture solver · Navier–Stokes mixture rotation | Loss α: 0.355 ± 0.005 -> 0.351 ± 0.005 | Helped |
| Apply rotation to the test data | FluxGNN · Navier–Stokes mixture test and rotation | Loss α: 0.349 ± 0.005 -> 0.351 ± 0.005 | No clear effect |
| Apply scaling to the test data | FluxGNN · Navier–Stokes mixture test and scaling | Loss α: 0.349 ± 0.005 -> 0.356 ± 0.005 | No clear effect |
| Evaluate on taller shapes not seen during training | FluxGNN · Navier–Stokes mixture taller | Loss α: 0.337 ± 0.006 vs PENN 0.357 ± 0.006 | Helped |
| Evaluate on taller shapes not seen during training | PENN · Navier–Stokes mixture taller | Loss α: test: 0.358 ± 0.005; taller: 0.357 ± 0.006 | Hurt |
Surprising findings
- The plain FVM baseline diverges (NaN) on the taller dataset for all tested settings, so the learned model's advantage comes from combining learned flux with FVM structure.
- Conservation is argued to be essential for generalization: PENN, despite E(n)-equivariance, degrades on the taller dataset and shows spurious velocity.
Limitations
- Long-term rollout is unstable; FluxGNN shows higher loss than MP-PDE after t approximately 7.0.
- No performance comparison against well-optimized classical solvers.
- Experiments are limited to flow and transport problems with small 2D domains and coarse resolution.
Experimental setup
Datasets: convection-diffusion trajectories generated from exact solution Navier-Stokes with mixture trajectories generated with OpenFOAM rotation, scaling, and taller out-of-domain test sets
Benchmarks: convection-diffusion equation test set Navier-Stokes with mixture test, rotation, scaling, taller datasets
Metrics: MSE loss RMSE loss conservation error speed-accuracy tradeoff
Baselines: MP-PDE PENN FVM
Builds on: finite volume method message passing neural networks MP-PDE PENN temporal bundling
Closest papers by meaning
- Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions (NeurIPS 2022 · similarity 0.93)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.90)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.90)
- Mechanistic PDE Networks for Discovery of Governing Equations (ICML 2025 · similarity 0.89)
- RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.88)
- P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics (NeurIPS 2024 · similarity 0.88)
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