Tensor Field Models

Alexander Strunk; Roland Assam

cs.LGAug 19, 2026

Tensor Field Models

Authors: Alexander Strunk, Roland Assam

Organizations: Evercot AI

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.

Figures & tables

Figure 1: A Tensor Field Model can be visualized as a collection of blue component Fields on latent Bundles. Through the Operator

\mathfrak{F}_{\phi}

, each admissible Field collection defines a red time-dependent tangent section on the Generative State Manifold

\mathcal{M}

. This supports batched or parallel evaluation of independent queries while reusing the same Field collection.

Panel of Figure 2 — Figure 2: Point-evaluation formulation of a component-separable constructed Tensor Field Model. The component constructors produce $\mathbf{H}_{c}=E_{\theta}(c)$ . For $N_{q}$ queries $q_{j}=(x_{j},t_{j})$ sharing the condition $c$ , the point-evaluation map $F_{\phi}$ reuses this collection to return $\widehat{v}_{c}(x_{j},t_{j})\in T_{x_{j}}\mathcal{M}$ . The state $x_{j}$ remains an evaluation argument. In a dynamical specialization, each trajectory can therefore be evaluated at its current state.

Figure 3: A Tensor Bundle induced by a carrier Bundle

C\to M

. The lower region represents the Base Manifold

M

, and the regions above selected base points represent Fibers

\mathbb{T}^{r}_{s}(C)_{x}

. A smooth section assigns to every

x\in M

one Tensor

A(x)

in the corresponding Fiber. The carrier Fibers

C_{x}

need not be tangent spaces of

M

Setting	Metric	Structured TFM	Control	Paired reduction
Bilinear $\mathbb{R}^{4}$	sliced $W_{2}$	$0.0549\,[0.0480,0.0617]$	$0.447\,[0.116,0.779]$	$82.9\%\,[68.6\%,97.3\%]$
IID $\mathbb{R}^{4}$ control	sliced $W_{2}$	$2.022\,[1.599,2.446]$	$1.944\,[1.610,2.278]$	$-4.1\%\,[-15.6\%,7.5\%]$
Haar-rotated $\mathbb{S}^{2}$	velocity NMSE	$0.2983\,[0.2946,0.3020]$	$0.3171\,[0.3100,0.3241]$	$5.92\%\,[4.94\%,6.89\%]$

Table 1: Primary held-out results. Model entries are mean

[95\%\text{ Student-}t\text{ CI}]

across five seeds. The last column is the mean seedwise percentage reduction of the TFM relative to its named control. The Euclidean control is direct FM. The spherical control is the Haar-augmented projected MLP. Lower is better.

Figure 4: Controlled experimental summary. (a) Seedwise held-out sliced

W_{2}

. Gray segments pair the same seed and black bars mark medians. The CS-TFM advantage on the bilinear family disappears for the IID control. (b) Canonical and jointly Haar-rotated velocity NMSE. All models are tangent. Pale lines are individual seeds and heavy lines are medians. The rotated raw inputs lie outside the sampled canonical training support of both the CS-TB-TFM and the unaugmented coordinate MLP. In the CS-TB-TFM, rotation leaves the coefficient inputs invariant and transforms the basis Fields covariantly. The augmented MLP sees such rotations during training. (c) Median serial CPU latency divided by cached-TFM latency. Rebuilding the identical TFM or recomputing the direct predictor is costly, but a generic cached graph and an exactly partially cached MLP match the TFM, showing that reuse is not unique to the terminology.

$B$	TFM serial ms	TFM batched ms	TFM speedup	Direct serial ms	Direct batched ms	Direct speedup
1	$6.57\,[5.63,8.23]$	$5.83\,[5.61,8.01]$	$1.13\times$	$3.99\,[3.91,4.15]$	$4.04\,[3.87,4.37]$	$0.99\times$
8	$52.06\,[47.37,54.71]$	$7.88\,[7.16,8.93]$	$6.61\times$	$32.20\,[31.58,33.10]$	$5.04\,[4.99,5.27]$	$6.39\times$
32	$242.72\,[215.22,311.89]$	$9.60\,[8.69,13.43]$	$25.28\times$	$129.13\,[128.45,129.85]$	$6.34\,[6.25,6.56]$	$20.38\times$
128	$1082.72\,[840.44,1161.49]$	$16.32\,[14.21,23.34]$	$66.33\times$	$528.81\,[520.16,535.19]$	$10.41\,[10.29,10.57]$	$50.82\times$
512	$2923.10\,[2880.50,3010.21]$	$25.92\,[25.50,27.38]$	$112.76\times$	$2103.91\,[2069.18,2165.00]$	$20.75\,[19.72,23.60]$	$101.41\times$

Table 2: CPU latency for independent-path sampling. Times are median

[\mathrm{IQR}]

in milliseconds. Speedup is the ratio of serial and batched medians within each model.

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Open paper AskPDF arXiv

We introduce a continuous-time generative modeling framework, motivated by the Chow-Rashevskii theorem, that builds expressive flows from a small set of fixed vector fields and learned scalar controls. Instead of learning an unconstrained high-dimensional vector field, our framework constructs the velocity by modulating fixed vector fields with learned scalar control functions. When the fixed fields are bracket-generating, their Lie algebra spans the ambient space, providing a mechanism for expressive transport with only a small number of learned control channels and offering a parameter-efficient geometric alternative to standard vector-field parameterizations. This decoupled formulation yields a structured and interpretable generative model in which the number of learned scalar output channels can be chosen independently of the ambient dimension. We formulate an expressivity principle showing that, under suitable controllability and well-posedness assumptions, such controlled flows can transport a source distribution to a target distribution. We train the resulting model using a continuous-normalizing-flow likelihood objective and present proof-of-concept experiments on synthetic distributions.

Abstract

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