Donut: A Multifaceted Scientific Signifier

• Donut is a term with versatile meanings encompassing annular geometries and compact integrated systems across various scientific domains.
• Its applications range from visual document understanding transformers, physics-informed diffraction networks, to anomaly detection in time-series data.
• Methodologies include mathematical modeling in discrete geometry, optical vortex beam design, and system integrations in machine learning and robotics.

Searching arXiv for recent and foundational papers on “DONUT” across domains to ground the article in published work. The term DONUT and its lowercase variant Donut are used across contemporary research to denote several distinct objects: annular visual encodings, beam and point-spread-function designs, physical effects, machine-learning systems, synthetic benchmarks, and bibliographic infrastructures. In the literature, the same label identifies an OCR-free document model, a variational anomaly detector, a physics-aware diffraction network, a decoder-only trajectory forecaster, a DNS fingerprinting tool, a synthetic 3D-topology benchmark, and multiple non-acronymic “donut” structures in geometry and physics (Kim et al., 2021, Xu et al., 2018, Luo et al., 18 Jul 2025, Knoche et al., 7 Jun 2025).

1. Terminological scope and acronymic reuse

In acronymic usage, DONUT is not a single framework but a recurrent naming pattern. The expansions documented in the literature span document understanding, diffraction analysis, trajectory prediction, software fingerprinting, topology benchmarking, and topology bibliometrics.

Usage	Expansion or meaning	Domain
Donut	Document understanding transformer	Visual document understanding
DONUT	Diffraction with Optics for Nanobeam by Unsupervised Training	X-ray nanodiffraction
DONUT	Decoder-Only Network for Unrolling Trajectories	Motion forecasting
DONUT	Domain Oriented Network Unmasking Tool	DNS-based software fingerprinting
DONUT	Dataset Of maNifold strUcTures	3D topology benchmark
DONUT	Database of Original Non-Theoretical Uses of Topology	TDA bibliography

These acronymic forms coexist with descriptive uses in which “donut” refers to a ring-like structure or annular pattern: a concentric-ring visualization for Spatial Social Networks, the vortex excitation beam in MINFLUX, the ring-like angular distribution in proton channeling, annular supergranular flow structures on the Sun, and a rectangle-with-hole construction in discrete geometry (Sarkar et al., 2021, Liu et al., 2024, Borka et al., 2010, Roudier et al., 2023, Murawski et al., 2024).

2. Geometric abstractions: concentric summaries and mathematical donuts

In Spatial Social Networks, the donut is a summary glyph rather than a node-link drawing. Sarkar and Yadav define it as a concentric-ring visualization centered at the viewport centroid, partitioned into eight equal angular sectors and three concentric distance buckets—Near, Medium, and Far—so that each sector-ring cell encodes the count of edges by direction and scale (Sarkar et al., 2021). The construction aggregates the visible network into an $8\times3$ matrix $Count[d,b]$, annotates the center with the number of nodes, and uses fixed sector angles $\Delta \theta = 2\pi/8 = \pi/4$ with ring radii $R_b = (b/3)\cdot R_{\max}$. The same routine is defined for both a network-level donut, using all nodes and edges in the current map extent, and a regional-level donut, recomputed on a spatial subset. Its stated advantages are simultaneous encoding of topology and spatial structure, multi-scale reuse, and avoidance of “hairball” clutter; its stated limitations are loss of individual-edge detail, dependence on sector and bucket design, and the burden of color-only magnitude encoding (Sarkar et al., 2021).

A different formalization appears in discrete geometry. A mathematical donut is an ordered quadruple $D=(a,b,x,y)$ of positive integers such that the outer rectangle of side lengths $a,b$ contains an inner rectangle of side lengths $x,y$, with the area constraint

$ab=2xy.$

The hole is twistable precisely when the $90^\circ$-rotated inner rectangle also fits, and the paper gives the necessary and sufficient criterion $2x>a$ and $Count[d,b]$0 (Murawski et al., 2024). For square donuts $Count[d,b]$1, existence is equivalent to $Count[d,b]$2 being the sum of the three positive entries of some Pythagorean triple, with the complete parametrization

$Count[d,b]$3

for $Count[d,b]$4, $Count[d,b]$5, and $Count[d,b]$6 not both odd (Murawski et al., 2024). This places the “donut” in an arithmetic setting rather than a radial one.

3. Annular patterns in optics, imaging, channeling, and solar dynamics

In MINFLUX localization microscopy, the donut is the conventional vortex excitation beam. It is obtained by choosing a pupil-plane phase $Count[d,b]$7, yielding an intensity PSF proportional to

$Count[d,b]$8

with a central zero and a first Bessel-ring pattern (Liu et al., 2024). Under equal-shape constraints on all excitation beams, numerical optimization with 15 Zernike modes and many random restarts converges to the donut, and the theoretical analysis identifies the vortex mask $Count[d,b]$9 as the unique global maximizer of the isotropic gradient-norm criterion (Liu et al., 2024). Quantitatively, for $\Delta \theta = 2\pi/8 = \pi/4$0, $\Delta \theta = 2\pi/8 = \pi/4$1, $\Delta \theta = 2\pi/8 = \pi/4$2, $\Delta \theta = 2\pi/8 = \pi/4$3 photons, and background $\Delta \theta = 2\pi/8 = \pi/4$4, donut-only MINFLUX with $\Delta \theta = 2\pi/8 = \pi/4$5 translated copies achieves $\Delta \theta = 2\pi/8 = \pi/4$6 at the center and $\Delta \theta = 2\pi/8 = \pi/4$7–$\Delta \theta = 2\pi/8 = \pi/4$8 on the full field of view, whereas two pairs of half-moon beams reach $\Delta \theta = 2\pi/8 = \pi/4$9 centrally and $R_b = (b/3)\cdot R_{\max}$0 on average, corresponding to an approximately $R_b = (b/3)\cdot R_{\max}$1 improvement in $R_b = (b/3)\cdot R_{\max}$2 and a reduction from $R_b = (b/3)\cdot R_{\max}$3 to $R_b = (b/3)\cdot R_{\max}$4 photons for $R_b = (b/3)\cdot R_{\max}$5 precision (Liu et al., 2024).

A related annular construction appears in phase-based stimulated emission depletion magnetic particle imaging. There, a spatially varying relaxation time induces a nonlinear phase lag $R_b = (b/3)\cdot R_{\max}$6, and the imaginary part of the harmonic PSF has a central null: $R_b = (b/3)\cdot R_{\max}$7 Subtracting this donut-shaped focal spot from the Lorentzian focal spot defines

$R_b = (b/3)\cdot R_{\max}$8

and the reported focal-spot size reduction is up to $R_b = (b/3)\cdot R_{\max}$9 beyond the Langevin magnetization resolution barrier (Jia et al., 10 May 2025).

In proton channeling through an $D=(a,b,x,y)$0 single-wall carbon nanotube, the donut is a ring-like angular distribution that develops as the proton incident angle $D=(a,b,x,y)$1 approaches the critical channeling angle $D=(a,b,x,y)$2 (Borka et al., 2010). The paper identifies the effect with rainbow scattering, governed by singularities of the mapping from entrance-plane coordinates to scattering angles, i.e. the vanishing of the Jacobian $D=(a,b,x,y)$3. Numerically, $D=(a,b,x,y)$4 with the image interaction and $D=(a,b,x,y)$5 without it; at $D=(a,b,x,y)$6, approximately $D=(a,b,x,y)$7, a full ring of high yield appears at radius $D=(a,b,x,y)$8 (Borka et al., 2010).

Solar-physics usage is again descriptive. In 6–12 h temporal averages of the photospheric horizontal-flow modulus, Roudier et al. identify nearly circular rings of enhanced speed and call each such ring a donut (Roudier et al., 2023). The azimuthally averaged profile is modeled as

$D=(a,b,x,y)$9

with $a,b$0 and $a,b$1 (Roudier et al., 2023). Reported radii are $a,b$2–$a,b$3, lifetimes range from $a,b$4 to $a,b$5 with a peak around $a,b$6, and the structures are largely absent from magnetized plage regions (Roudier et al., 2023). The paper interprets them as the most active convective cells associated with supergranulation and shows, through cork advection, that the strongest donuts alone can account for much of quiet-Sun magnetic-flux diffusion (Roudier et al., 2023).

4. Donut and DONUT as machine-learning models

In visual document understanding, Donut denotes the Document understanding transformer, an OCR-free encoder-decoder model that maps a raw document image directly to a target token sequence convertible to JSON (Kim et al., 2021). Its visual encoder is Swin-Transformer-B with stage depths $a,b$7, hidden dimensions $a,b$8, and attention heads $a,b$9, while the textual decoder is a 4-layer BART-style cross-attention decoder with $x,y$0, $x,y$1, $x,y$2, and learned positional embeddings for up to 1,536 decoding steps (Kim et al., 2021). Pre-training is cast as text reading with cross-entropy

$x,y$3

and SynthDoG contributes approximately $x,y$4 M synthetic pages per language; combined with $x,y$5 M IIT-CDIP English scans, the pre-training corpus comprises $x,y$6 synthetics plus $x,y$7 reals (Kim et al., 2021). Reported downstream results include $x,y$8 on RVL-CDIP at $x,y$9 per image, CORD $ab=2xy.$0 F1 and $ab=2xy.$1 TED-Acc at $ab=2xy.$2, Ticket $ab=2xy.$3 F1 and $ab=2xy.$4 TED-Acc at $ab=2xy.$5, and DocVQA $ab=2xy.$6 ANLS, with $ab=2xy.$7 on the handwritten subset (Kim et al., 2021). A domain application to construction specification tables of contents fine-tunes a public Donut base model on 200 annotated pages, using 180 for fine-tuning and 20 for testing, and reports field-wise accuracies $ab=2xy.$8 for heading number, $ab=2xy.$9 for heading title, $90^\circ$0 for subheading number, $90^\circ$1 for subheading title, and $90^\circ$2 on average (Feyisa et al., 2024). A later compression study analyzes Donut’s decoder circuits, identifies $90^\circ$3 as the sole locus of transcription and $90^\circ$4 as negligible, and derives Donut-MINT variants at $90^\circ$5, $90^\circ$6, and $90^\circ$7 decoder budgets; on DocVQA, the $90^\circ$8 variant reports $90^\circ$9 ANLS, $2x>a$0 EM, $2x>a$1 M parameters, and $2x>a$2 latency, versus $2x>a$3 ANLS, $2x>a$4 EM, 257 M parameters, and $2x>a$5 for the teacher (Mansour et al., 30 Sep 2025).

In time-series anomaly detection, Donut is an unsupervised VAE for seasonal KPIs in web applications (Xu et al., 2018). The model uses fully connected encoder and decoder networks with two hidden layers of 100 ReLU units each, a Gaussian latent prior $2x>a$6, and sliding windows of length $2x>a$7 (Xu et al., 2018). Its central methodological additions are a modified ELBO that de-weights missing and anomalous points,

$2x>a$8

missing-data injection with $2x>a$9, and MCMC-based imputation with $Count[d,b]$00 steps at inference (Xu et al., 2018). On three KPI datasets, its best F-scores range from $Count[d,b]$01 to $Count[d,b]$02, outperforming both a supervised ensemble baseline and a baseline VAE; the paper also introduces a KDE interpretation of reconstruction probability (Xu et al., 2018).

In X-ray science, DONUT means Diffraction with Optics for Nanobeam by Unsupervised Training, a physics-aware autoencoder for scanning X-ray nanodiffraction microscopy (Luo et al., 18 Jul 2025). A 2D CNN encoder maps each $Count[d,b]$03 diffraction image to a latent vector $Count[d,b]$04, representing relative lattice strain, in-plane tilt, and out-of-plane tilt, while two heads consume $Count[d,b]$05: a mirror-symmetric CNN decoder $Count[d,b]$06 and a fully differentiable forward scattering model $Count[d,b]$07 (Luo et al., 18 Jul 2025). Training uses only the raw diffraction images through the composite loss

$Count[d,b]$08

with $Count[d,b]$09 and $Count[d,b]$10 (Luo et al., 18 Jul 2025). The simulated dataset spans a $Count[d,b]$11 grid over $Count[d,b]$12 for 68,921 diffraction patterns, while the experimental dataset is one $Count[d,b]$13 scan with approximately 27,000 patterns (Luo et al., 18 Jul 2025). Inference with the encoder alone runs at approximately $Count[d,b]$14 on GPU and $Count[d,b]$15 on CPU, compared with $Count[d,b]$16 for conventional fitting, and experimental agreement with correlation-library fitting is summarized by Pearson $Count[d,b]$17, $Count[d,b]$18, and $Count[d,b]$19 (Luo et al., 18 Jul 2025).

5. Forecasting, speech, telemetry, and robotic manipulation

DONUT also names a decoder-only trajectory forecaster. Decoder-Only Network for Unrolling Trajectories tokenizes trajectories into sub-trajectories of $Count[d,b]$20 time steps, embeds them in dimension $Count[d,b]$21, and autoregressively predicts future motion without a separate encoder (Knoche et al., 7 Jun 2025). Its factorization is

$Count[d,b]$22

and training adds an auxiliary overprediction branch with horizon $Count[d,b]$23 (Knoche et al., 7 Jun 2025). On the Argoverse 2 single-agent benchmark, the reported hidden-test performance is $Count[d,b]$24-minFDE$Count[d,b]$25, minFDE$Count[d,b]$26, minADE$Count[d,b]$27, and MR$Count[d,b]$28 (Knoche et al., 7 Jun 2025).

In speech technology, DONUT is a CTC-based query-by-example wakeword detector for personalized keyword spotting (Lugosch et al., 2018). Enrollment uses three user-recorded examples, a pre-trained 3-layer unidirectional GRU label model, beam search to generate phoneme-sequence hypotheses, and weighted aggregation of CTC forward scores during inference (Lugosch et al., 2018). The model size is 168 k parameters. On English-Fewshot, reported EERs are $Count[d,b]$29, $Count[d,b]$30, and $Count[d,b]$31 across the three negative conditions shown in Table 1, versus $Count[d,b]$32, $Count[d,b]$33, and $Count[d,b]$34 for DTW on FBANK features (Lugosch et al., 2018).

In network telemetry, DONUT denotes the Domain Oriented Network Unmasking Tool, a rule-based system for DNS-based software fingerprinting (Schäfer et al., 2022). It processes passively monitored DNS traffic through a parser, a packet-level CF-Matcher, and a host-level Set-Matcher mediated by an IP dictionary (Schäfer et al., 2022). CF-rules match domain patterns and query types to labels, while Set-rules infer applications from required and optional label sets. On the performance dataset, DONUT processes approximately 15,000 packets per second, corresponding to 1,000,000 packets in 67 s; on a four-VM validation set with optimized parameters, it achieved zero false positives and zero false negatives, giving precision, recall, and F1 approximately $Count[d,b]$35 (Schäfer et al., 2022).

In robotics, “Make a Donut” names a zero-shot deformable-manipulation system that uses a LLM for high-level stage decomposition and EMD-space planning for low-level control (You et al., 2023). For the donut task, the LLM specifies stages such as flattening dough into a disc, shaping a ring, and punching the hole, along with Python code that generates subgoal point clouds (You et al., 2023). The low-level planner iteratively applies gradient steps in Earth Mover’s Distance space,

$Count[d,b]$36

and then optimizes tool actions through differentiable physics with a point-to-point loss $Count[d,b]$37 (You et al., 2023). Reported donut performance is a normalized EMD-decrease score of $Count[d,b]$38 with $Count[d,b]$39 success, and the paper states that the method surpasses multiple baselines in dough manipulation without demonstrations (You et al., 2023).

6. DONUT as benchmark and bibliographic infrastructure for topology

Two further usages shift from algorithms to research infrastructure. In geometric deep learning, DONUT is the Dataset Of maNifold strUcTures, introduced to test whether 3D point-cloud encoders capture global topology (Martinez et al., 24 Apr 2026). The benchmark contains 29,517 distinct 3D manifold meshes with $Count[d,b]$40 and total genus $Count[d,b]$41, sampled nearly uniformly over that grid (Martinez et al., 24 Apr 2026). Each mesh is built from superquadrics, $Count[d,b]$42-tori, or cones; 1,024 surface points are sampled and normalized to the unit sphere, and ground-truth $Count[d,b]$43 persistence diagrams are computed with GUDHI v3.11.0 using an $Count[d,b]$44-filtration, հետո thresholded to retain the top 10% most persistent pairs (Martinez et al., 24 Apr 2026). On the DONUT test split of 5,938 shapes, FILTR predicts persistence diagrams with best reported $Count[d,b]$45, $Count[d,b]$46, and PIE $Count[d,b]$47, while linear probes on frozen pretrained encoders achieve only modest performance on $Count[d,b]$48 and $Count[d,b]$49 prediction, with best values of approximately $Count[d,b]$50 and $Count[d,b]$51, respectively (Martinez et al., 24 Apr 2026).

In topological data analysis bibliometrics, DONUT means Database of Original Non-Theoretical Uses of Topology, a curated database of papers on practical TDA applications (Giunti et al., 2023). The project began from a 2017 group chat associated with the HIM Spring School and later migrated from a shared Google Sheet to a Zotero library and then to a public search interface (Giunti et al., 2023). By late 2022, the catalog had grown to 431 entries, of which 58 were labeled innovate (Giunti et al., 2023). Each entry carries at least one tag in each of three classes—area of application, mathematical tools, and input type—and an optional flavor label such as confirm or innovate (Giunti et al., 2023). The maintenance pipeline exports BibTeX through Pyzotero, indexes it with Xapian, and serves it through a Flask web front end (Giunti et al., 2023).

7. Cross-disciplinary regularities

Across these usages, no single technical definition of DONUT is canonical. Instead, the name repeatedly marks either an annular geometry or a compact, end-to-end system with a domain-specific acronym. The annular cases emphasize central-null or ring-shaped structure—concentric sector summaries in Spatial Social Networks, vortex and depletion PSFs in microscopy and magnetic particle imaging, rainbow rings in channeling, and annular supergranular flows on the Sun (Sarkar et al., 2021, Liu et al., 2024, Jia et al., 10 May 2025, Borka et al., 2010, Roudier et al., 2023). The acronymic cases emphasize integrated pipelines: direct pixels-to-JSON document understanding, unsupervised seasonal-KPI anomaly detection, physics-aware diffraction inference, decoder-only motion forecasting, rule-based DNS software inference, and curated topology resources (Kim et al., 2021, Xu et al., 2018, Luo et al., 18 Jul 2025, Knoche et al., 7 Jun 2025, Schäfer et al., 2022, Giunti et al., 2023).

This pattern suggests that “DONUT” functions less as a stable concept than as a reusable scientific signifier. In some fields it denotes literal ring structure; in others it denotes methodological compactness, architectural closure, or a memorable acronym. A plausible implication is that the term persists because it accommodates both visually intuitive geometry and concise naming, allowing unrelated communities to adopt it without semantic conflict.