Equivariant atomistic machine learning models have largely been built on spherical-tensor representations, where explicit angular-momentum coupling introduces substantial complexity and systematic extensions beyond energies and forces remain challenging, often requires problem-specific architectural choices. Here we introduce the Tensor Atomic Cluster Expansion (TACE), which unifies scalar and tensorial modeling in Cartesian and space by decomposing local environments into irreducible Cartesian tensors (ICT) constructing a controlled many-body hierarchy with atomic cluster expansion (ACE). In addition to performing ACE in the frequency domain, we propose an efficient Clebsch-Gordan-free alternative in the spatial domain. TACE provides universal invariant (e.g., fidelity tags and charges) and equivariant (e.g., external electric fields and non-collinear magnetic moments) embeddings and predicted tensorial observables are handled on equal footing and enabling explicit control at inference. We demonstrate the accuracy, stability, and efficiency across finite molecules and extended materials, including in-domain and out-of-domain benchmarks, spectra, Hessian, external-field responses, charged systems, and multi-fidelity/head training. We further show its robustness on nonequilibrium/reactive datasets and controlled scaling when extending to large foundation model datasets.

翻译：等变原子机器学习模型主要建立在球张量表示基础上，其中显式的角动量耦合引入了显著的复杂性，且超越能量和力的系统性扩展仍具挑战性，通常需要针对特定问题的架构选择。本文引入张量原子簇展开（TACE），该方法通过在笛卡尔空间中分解局域环境为不可约笛卡尔张量（ICT），并结合原子簇展开（ACE）构建受控的多体层级，从而统一了标量与张量建模。除了在频域执行ACE外，我们提出了一种在空域中高效的无Clebsch-Gordan替代方案。TACE提供了普适的不变量（如保真度标签和电荷）与等变量（如外电场和非共线磁矩）嵌入，且预测的张量观测量可在同等基础上处理，并支持在推理阶段进行显式控制。我们在有限分子和扩展材料体系上展示了其准确性、稳定性和效率，包括域内和域外基准测试、光谱、Hessian矩阵、外场响应、带电系统以及多保真度/多头训练。我们进一步证明了其在非平衡/反应数据集上的鲁棒性，以及在扩展至大型基础模型数据集时受控的缩放特性。