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Equivariant Matrix Function Neural Networks
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Introduces Matrix Function Neural Networks (MFNs), which parameterize learnable equivariant graph operators and apply learnable resolvent (pole-expansion) matrix functions to capture non-local interactions, evaluated on cumulene quantum chemistry extrapolation, ZINC, and TU datasets.
Problem
Message-passing GNNs and standard spectral GNNs struggle to model non-local many-body interactions (e.g., in conjugated molecules, metals, amorphous materials) and are often restricted to pure graphs without geometric/equivariant symmetries.
Approach
Learn equivariant self-adjoint graph matrices from local MACE node features and apply a matrix function parameterized by a resolvent pole expansion with learnable poles and weights, giving non-local equivariant features whose cost could scale linearly with sparse selected-inversion methods.
Key results
- On ZINC, MFN (GCN) reaches 0.063 MAE, beating the next best PDF (0.066) with ~512k parameters.
- On the GNL cumulene test set (out-domain 15,16 carbons), MFN gets 2.6 meV/atom energy RMSE and 37.7 meV/A force RMSE versus MACE's 12.0 and 77.1 (energy worse for MACE but the table bolds MFN).
- On TU datasets, MFN (GCN) is best on MUTAG (91.5) and PTC_MR (68.9) and competitive on other subsets; it does not top IMDB-B (74.1 vs PDF 75.6).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MFN (MACE) instead of local MACE | MACE · Cumulene out-of-domain test (11,12 and 15,16 carbon atoms) | energy and force RMSE: energy: 16.5 and 12.0 -> 0.9 and 2.6; forces: 108.5 and 77.1 -> 22.5 and 37.7 | Helped |
| Use MFN (MACE) instead of SpookyNet global attention | SpookyNet · Cumulene train and in-domain test | energy RMSE: 31.4 -> 2.0; 30.8 -> 2.6 | Helped |
| Use MFN (MACE) instead of SpookyNet global attention | SpookyNet · Cumulene train and in-domain test | force RMSE: 114.1 -> 31.7; 162.3 -> 34.0 | Helped |
| Use global attention instead of local MACE | Cumulene models · Cumulene train and in-domain test | energy RMSE: 41.1 -> 31.4; 41.8 -> 30.8 | Helped |
| Use global attention instead of local MACE | Cumulene models · Cumulene out-of-domain test (11,12 and 15,16 carbon atoms) | energy RMSE: 16.5 -> 31.4; 12.0 -> 23.4 | Hurt |
| Use global attention instead of local MACE | Cumulene models · Cumulene out-of-domain test (11,12 and 15,16 carbon atoms) | force RMSE: 108.5 -> 116.2; 77.1 -> 87.6 | Hurt |
| Use MFN (GCN) instead of PDF | ZINC graph prediction · ZINC | MAE: 0.066$\pm$0.004 -> 0.063$\pm$0.002 | Helped |
| Use MFN (GCN) instead of PDF | MUTAG graph classification · MUTAG | accuracy: 89.91$\pm$4.35 -> 91.5$\pm$7.35 | Helped |
| Use MFN (GCN) instead of PDF | ENZYMES graph classification · ENZYMES | accuracy: 73.50$\pm$6.39 -> 72.9$\pm$7.55 | Hurt |
| Use MFN (GCN) instead of PDF | PTC_MR graph classification · PTC_MR | accuracy: 68.36$\pm$8.38 -> 68.9$\pm$8.09 | Helped |
| Use MFN (GCN) instead of PPGNN | PROTEINS graph classification · PROTEINS | accuracy: 77.20$\pm$4.73 -> 76.18$\pm$4.07 | Hurt |
| Use MFN (GCN) instead of PDF | IMDB-B graph classification · IMDB-B | accuracy: 75.60$\pm$2.69 -> 74.1$\pm$1.04 | Hurt |
Surprising findings
- Global attention (SpookyNet) fails to extrapolate to longer cumulene chains, performing worse than a simple local MACE model on out-of-domain sizes.
- The current MFN prototype has cubic scaling and is about 4x slower than MACE per atom in the reported runtime comparison.
Limitations
- Current implementation has cubic scaling; linear scaling requires selected inversion and sparse kernels not implemented.
- Evaluation of non-locality is limited mainly to cumulene chains and a small GNL dataset.
- Matrix function evaluation is slower than MACE in runtime comparison.
Experimental setup
Datasets: ZINC MUTAG ENZYMES PTC-MR PROTEINS IMDB-B Guaranteed Non-Local (GNL) cumulene dataset
Benchmarks: ZINC MUTAG ENZYMES PTC-MR PROTEINS IMDB-B GNL cumulene dataset
Metrics: MAE accuracy energy RMSE (meV/atom) force RMSE (meV/A) runtime (ms/atom)
Baselines: MACE SpookyNet GCN GAT MPNN GT SAN Graphormer PDF GIN
Builds on: MACE Spectral GNN GPRGNN PEXSI SpookyNet
Closest papers by meaning
- On the Generalization of Equivariant Graph Neural Networks (ICML 2024 · similarity 0.87)
- Universally Invariant Learning in Equivariant GNNs (NeurIPS 2025 · similarity 0.86)
- On the Expressive Power of Sparse Geometric MPNNs (ICLR 2025 · similarity 0.86)
- Learning the Electronic Hamiltonian of Large Atomic Structures (ICML 2025 · similarity 0.86)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.86)
- Equivariant Polynomial Functional Networks (ICML 2025 · similarity 0.85)
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