arXiv:2605. 20440v2 Announce Type: replace Abstract: Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude.
arXiv:2409. 15600v3 Announce Type: replace Abstract: A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general.
arXiv:2608.18714v2 Announce Type: replace-cross Abstract: Crystal symmetry dictates whether a physical response tensor must vanish, establishing a direct test for machine learning predictions indepen...
arXiv:2607. 06634v1 Announce Type: new Abstract: Compact networks built from Clifford algebra Cl(3,0) primitives are exactly SO(3)-equivariant and learn synthetic 3D vector laws from few samples.
The article demonstrates that a single design decision—whether a machine‑learning model’s features include parity labels—determines if the model can ever predict physically impossible values for material properties. Using group‑theoretical analysis, the authors introduce the parity gap criterion to identify which properties and crystal symmetries are affected. Experiments on 2,000 centrosymmetric crystals show that parity‑labelled models achieve exact zero predictions for forbidden piezoelectric responses, whereas models lacking parity labels produce large errors, yet both maintain comparable overall accuracy.
arXiv:2606. 29584v1 Announce Type: cross Abstract: $\mathrm{Cl}(3,0)$ interatomic potentials, despite their algebraic elegance, predict force magnitudes accurately but force directions poorly.
The paper addresses the limitations of spectral dimension reduction for dynamical systems composed of weakly interacting components, where standard rank‑based methods either require exponentially many modes or omit entire components (a phenomenon termed linear masking). It proposes scoring the σ‑algebra generated by coordinates instead of individual modes, using a χ²‑divergence criterion that guarantees an embedding with twice the intrinsic dimension captures the full operator spectrum. Experiments on benchmark systems show that this algebraic approach recovers masked components and enables accurate prediction from few labels, outperforming traditional rank‑based and VAMP methods.
arXiv:2605. 18106v3 Announce Type: replace-cross Abstract: A striking geometric disparity has long persisted in the practice of deep learning.
The paper proposes a new dimension‑reduction strategy for transfer‑operator models of dynamical systems that focuses on scoring the σ‑algebra generated by coordinates rather than the operator’s spectral span. By using a χ²‑divergence criterion between embedded present and future states, the method guarantees that twice the intrinsic system dimension suffices to capture the full operator spectrum, even for systems with weakly interacting components that would otherwise require exponentially many modes. Experiments on benchmark systems show that this algebraic approach recovers masked components missed by rank‑based methods and enables accurate prediction of those components from few labels.
arXiv:2607. 03108v1 Announce Type: new Abstract: Post-hoc analysis of trained neural network weights often seeks to recover geometric structure directly from the parameters.
arXiv:2609.24942v1 Announce Type: new Abstract: A model generalizes outside its training distribution only when it computes a representation structurally equivalent to the generating mechanism, not a...
The paper introduces a new class of selective state space models (SSMs) that move beyond traditional diagonal constraints by leveraging exact non‑Abelian group tracking and solvable affine transformation groups. It reports significant performance gains, including a 0.04‑degree dead‑reckoning error, high accuracy on Dyck‑2 and deep AST scope tracking tasks, and demonstrates that strict isometry is essential for lossless long‑range associative memory.
arXiv:2609.07031v1 Announce Type: new Abstract: Learning with group invariances is central to many scientific and geometric learning problems, yet its computational foundations remain poorly understo...