TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction

Shi Yin, Xinyang Pan, Fengyan Wang, Lixin He
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:72364-72392, 2025.

Abstract

We propose a framework to combine strong non-linear expressiveness with strict SO(3)-equivariance in prediction of the electronic-structure Hamiltonian, by exploring the mathematical relationships between SO(3)-invariant and SO(3)-equivariant quantities and their representations. The proposed framework, called TraceGrad, first constructs theoretical SO(3)-invariant trace quantities derived from the Hamiltonian targets, and use these invariant quantities as supervisory labels to guide the learning of high-quality SO(3)-invariant features. Given that SO(3)-invariance is preserved under non-linear operations, the learning of invariant features can extensively utilize non-linear mappings, thereby fully capturing the non-linear patterns inherent in physical systems. Building on this, we propose a gradient-based mechanism to induce SO(3)-equivariant encodings of various degrees from the learned SO(3)-invariant features. This mechanism can incorporate powerful non-linear expressive capabilities into SO(3)-equivariant features with correspondence of physical dimensions to the regression targets, while theoretically preserving equivariant properties, establishing a strong foundation for predicting electronic-structure Hamiltonian. Experimental results on eight challenging benchmark databases demonstrate that our method achieves state-of-the-art performance in Hamiltonian prediction.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-yin25b, title = {{T}race{G}rad: a Framework Learning Expressive {SO}(3)-equivariant Non-linear Representations for Electronic-Structure {H}amiltonian Prediction}, author = {Yin, Shi and Pan, Xinyang and Wang, Fengyan and He, Lixin}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {72364--72392}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/yin25b/yin25b.pdf}, url = {https://proceedings.mlr.press/v267/yin25b.html}, abstract = {We propose a framework to combine strong non-linear expressiveness with strict SO(3)-equivariance in prediction of the electronic-structure Hamiltonian, by exploring the mathematical relationships between SO(3)-invariant and SO(3)-equivariant quantities and their representations. The proposed framework, called TraceGrad, first constructs theoretical SO(3)-invariant trace quantities derived from the Hamiltonian targets, and use these invariant quantities as supervisory labels to guide the learning of high-quality SO(3)-invariant features. Given that SO(3)-invariance is preserved under non-linear operations, the learning of invariant features can extensively utilize non-linear mappings, thereby fully capturing the non-linear patterns inherent in physical systems. Building on this, we propose a gradient-based mechanism to induce SO(3)-equivariant encodings of various degrees from the learned SO(3)-invariant features. This mechanism can incorporate powerful non-linear expressive capabilities into SO(3)-equivariant features with correspondence of physical dimensions to the regression targets, while theoretically preserving equivariant properties, establishing a strong foundation for predicting electronic-structure Hamiltonian. Experimental results on eight challenging benchmark databases demonstrate that our method achieves state-of-the-art performance in Hamiltonian prediction.} }
Endnote
%0 Conference Paper %T TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction %A Shi Yin %A Xinyang Pan %A Fengyan Wang %A Lixin He %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-yin25b %I PMLR %P 72364--72392 %U https://proceedings.mlr.press/v267/yin25b.html %V 267 %X We propose a framework to combine strong non-linear expressiveness with strict SO(3)-equivariance in prediction of the electronic-structure Hamiltonian, by exploring the mathematical relationships between SO(3)-invariant and SO(3)-equivariant quantities and their representations. The proposed framework, called TraceGrad, first constructs theoretical SO(3)-invariant trace quantities derived from the Hamiltonian targets, and use these invariant quantities as supervisory labels to guide the learning of high-quality SO(3)-invariant features. Given that SO(3)-invariance is preserved under non-linear operations, the learning of invariant features can extensively utilize non-linear mappings, thereby fully capturing the non-linear patterns inherent in physical systems. Building on this, we propose a gradient-based mechanism to induce SO(3)-equivariant encodings of various degrees from the learned SO(3)-invariant features. This mechanism can incorporate powerful non-linear expressive capabilities into SO(3)-equivariant features with correspondence of physical dimensions to the regression targets, while theoretically preserving equivariant properties, establishing a strong foundation for predicting electronic-structure Hamiltonian. Experimental results on eight challenging benchmark databases demonstrate that our method achieves state-of-the-art performance in Hamiltonian prediction.
APA
Yin, S., Pan, X., Wang, F. & He, L.. (2025). TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:72364-72392 Available from https://proceedings.mlr.press/v267/yin25b.html.

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