TFMs are introduced, realization-level Mathematical Structures in which a learned Operator maps a product of admissible component-section families to a prescribed family of time-dependent tangent sections on a Generative State Manifold.
Abstract
This paper introduces Tensor Field Models (TFMs), realization-level Mathematical Structures in which a learned Operator maps a product of admissible component-section families to a prescribed family of time-dependent tangent sections on a Generative State Manifold. Analytic and dynamical restrictions are encoded through the choice of admissible families rather than imposed by the root definition. Constructed, component-separable, and Tensor Bundle TFMs provide structured refinements of this common object. In the conditional realizations considered here, a structured condition $c=(c_1,\ldots,c_n)$ is mapped componentwise to a reusable collection $\mathbf H_c=(H_{c_1}^{(1)},\ldots,H_{c_n}^{(n)})$. In the architectures evaluated here, the component representations remain distinct and are combined only by the Field Operator to produce the generated Vector Field. All learned models are trained using Flow Matching. Experiments show that TFMs can improve performance and that amortized sampling enabled by reusable condition representations can accelerate generation.
This work provides a unique normal form and associated polynomial-time reduction strategy for tensor trains over arbitrary fields, and crucially provides the first tensor train form with the uniqueness property.
A framework for E(3)-equivariant UQ is introduced, modeling the full predictive distribution where both mean and covariance preserve rotational symmetry, and a Log-Euclidean Equivariant Scoring Objective (LE-ESO) is formulated, a robust surrogate loss based on the Multivariate Laplace distribution providing robustness to heavy-tailed errors and stable optimization.
Ruihan Liu, Yunting Ji, Jianbo Yu et al.· 0 citations
Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to $2^{70}$ configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.
Xiao Wang, Tomohiro Hashizume, Pia Siegl et al.· 2 citations
We develop and implement a positive tensor-network parameterization for computing Ricci-flat K\"ahler metrics on Calabi-Yau manifolds. It replaces the large Hermitian coefficient matrix of a high-degree algebraic metric by a matrix-product factorization. For the immersed source spaces used here, the resulting metric is globally positive for every parameter value and, at fixed local and bond dimensions, its number of parameters grows only linearly with the algebraic degree. We test the construction on three generalized complete-intersection Calabi-Yau (gCICY) threefolds, constructing chart by chart the generalized sections, holomorphic volume forms and sampling measures that define and train the metric there. From a common low-degree metric, the tensor network outperforms a parameter-matched neural potential using the same section data, reducing both bulk errors and the one-percent tail conditional mean in every paired run. It also reaches a substantially lower error than direct optimization of an unrestricted Hermitian metric of the same degree from the same start, with both methods optimized to validation convergence under their respective schedules. On a second geometry, a higher-degree network with fewer parameters than a lower-degree unrestricted Hermitian baseline substantially reduces the same-sample errors. We further observe saturation within the tested calculations: at fixed bond dimension, increasing the degree eventually plateaus; increasing the bond dimension at fixed optimization effort gives no resolved gain; and the outcome depends strongly on initialization and optimization path.
An efficient numerical approach for compressing a high-dimensional discrete distribution function into a non-negative tensor train (NTT) format and observing that the proposed NTT fitting procedure exhibits drastically faster convergence than an alternative multiplicative update method that has been previously proposed is observed.
Xun Tang, Rajat Vadiraj Dwaraknath, Le-Xing Ying· SIAM Journal on Scientific C...· 0 citations
The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.
V. Nazarenko, T. Lidzhiev, A. Tarakanov· arXiv.org· 0 citations
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