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Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing
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Proposes many-body equivariant message-passing potentials (ICTP) built on higher-rank irreducible Cartesian tensors and their products, adapting MACE to the Cartesian basis, and matches or beats spherical and Cartesian SOTA on several MLIP benchmarks.
Problem
Existing equivariant MLIPs rely on spherical tensors, whose Clebsch-Gordan products are costly, while state-of-the-art Cartesian models are restricted to invariant convolutions and limited many-body features.
Approach
Construct symmetric traceless irreducible Cartesian tensors of arbitrary rank from unit vectors and define irreducible Cartesian tensor products, then use them for equivariant convolutions and many-body (product-basis) features inside a MACE-style architecture (ICTP), with proofs of equivariance and tracelessness.
Key results
- On rMD17 with 950 training configurations, ICTP_sym reaches e.g. aspirin energy MAE 2.27 meV and force MAE 6.67 meV/Å, comparable to MACE (2.2 / 6.6).
- With 50 training configurations on rMD17, ICTP_sym outperforms MACE and NequIP on most molecules (e.g. aspirin energy 14.84 vs 17.0 meV for MACE).
- On Ta-V-Cr-W, ICTP_sym achieves 1.38 meV/atom energy error versus 2.19 meV/atom for MACE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use higher-rank irreducible Cartesian message passing rather than MACE | ICTP_sym · rMD17 with 50 training configurations | energy MAE: 17.0 -> 14.84 $\pm$ 0.98 | Helped |
| use higher-rank irreducible Cartesian message passing rather than MACE | ICTP models · 3BPA | energy and force RMSE: 2.70 $\pm$ 0.08 vs 3.0 $\pm$ 0.2; 9.39 $\pm$ 0.31 vs 8.8 $\pm$ 0.3 | Mixed |
| use higher-rank irreducible Cartesian message passing rather than MACE | ICTP_sym · Ta--V--Cr--W | energy error: 1.38 $\pm$ 0.09 vs 2.19 $\pm$ 0.31 | Helped |
| use higher-rank irreducible Cartesian message passing rather than MACE | ICTP_sym · Ta--V--Cr--W | force error: 0.028 $\pm$ 0.001 vs 0.029 $\pm$ 0.001 | No clear effect |
| replace the symmetric product basis with the full product basis (ablation) | ICTP_sym · 3BPA 300 K | energy RMSE: 2.70 $\pm$ 0.08 -> 2.70 $\pm$ 0.22 | No clear effect |
| construct the product basis in latent feature space (ablation) | ICTP_sym · 3BPA 300 K | inference time: 5.31 $\pm$ 0.02 -> 3.51 $\pm$ 0.22 | Helped |
| construct the product basis in latent feature space (ablation) | ICTP_sym · 3BPA 300 K | energy RMSE: 2.70 $\pm$ 0.08 -> 2.98 $\pm$ 0.34 | No clear effect |
| use the symmetric product basis rather than the full product basis (ablation) | ICTP · 3BPA 300 K | inference time: 6.45 $\pm$ 0.50 -> 5.31 $\pm$ 0.02 | Helped |
| use the symmetric product basis rather than the full product basis (ablation) | ICTP · Acetylacetone 300 K | energy RMSE: 0.75 $\pm$ 0.04 -> 0.76 $\pm$ 0.03 | No clear effect |
Surprising findings
- ICTP_full is about 1.4 times slower than MACE on 3BPA despite similar or slightly better accuracy, and ICTP inference is slower than MACE on the Ta-V-Cr-W data set due to the nu-fold product pre-factor.
- The authors could not reproduce MACE's originally reported 3BPA results with the current source code and had to re-run MACE under a unified setup.
Limitations
- Cartesian product cost grows factorially with rank L, so the approach is only advantageous for L <= 4 and the inference cost pre-factor from the nu-fold product basis is high.
- Only even tensor products are used to mirror MACE, so odd products are not exploited in the architecture.
Experimental setup
Datasets: rMD17 MD22 3BPA acetylacetone Ta-V-Cr-W
Benchmarks: rMD17 MD22 3BPA acetylacetone Ta-V-Cr-W
Metrics: energy MAE force MAE energy RMSE force RMSE inference time memory consumption
Baselines: MACE NequIP Allegro TensorNet CACE
Builds on: MACE NequIP Cartesian tensor products (Thomas et al.)
Closest papers by meaning
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