PolyGen: Generative Models in 3D, Polymers & Math

• The paper introduces an autoregressive Transformer model that factorizes 3D mesh generation into vertex and face components, achieving nearly 85% vertex and 90% face prediction accuracy with competitive reconstruction performance.
• polyGen’s framework for atomic-level polymer structure generation employs a diffusion-based Transformer to iteratively refine latent representations, yielding realistic polymer conformations that outperform rule-based methods.
• Mathematical PolyGen defines degenerate multi-poly-Genocchi polynomials via generating functions, enabling new combinatorial identities and analytical tools in special function theory.

PolyGen is a term used for distinct concepts within mathematical, statistical, algorithmic, and generative modeling domains. It most notably refers to (1) an autoregressive Transformer-based generative model for 3D meshes in graphics and robotics (Nash et al., 2020), (2) a novel learning-based framework for atomic-level synthetic polymer structure generation (Jain et al., 24 Apr 2025), and (3) several mathematical objects in special function theory related to poly-Genocchi polynomials (Kim et al., 2020, Corcino et al., 2020). This entry focuses on the principal instances where “PolyGen” denotes an algorithm or mathematical object of consequence.

1. PolyGen: Autoregressive Generative Model of 3D Meshes

“PolyGen: An Autoregressive Generative Model of 3D Meshes” (Nash et al., 2020) introduces a probabilistic, Transformer-driven framework for direct mesh generation. The approach factorizes the joint mesh distribution $p(\mathcal{M})$ into a vertex model $p(\mathcal{V})$ and a conditional face model $p(\mathcal{F}|\mathcal{V})$. Vertex sequences are ordered and represented as quantized coordinates, enabling categorical prediction at each time step. The architecture utilizes embedding schemes to encode coordinate type, sequence position, and quantized value.

Face connectivity is generated via an autoregressive pointer network: contextual vertex embeddings are encoded with a Transformer, and face selection is performed using dot-product attention over variable vertex indices, formalized as $$p(f_n = k | \cdots) = \mathrm{softmax}_k(p_n^\top e_k)$$.

PolyGen conditions mesh generation on diverse inputs, such as class labels, images, or voxel grids. Its flexible conditioning is implemented through context vector injection and cross-attention mechanisms in Transformer layers, facilitating adaptation to ambiguous or information-poor input modalities.

PolyGen’s probabilistic design allows explicit uncertainty representation in mesh structure, yielding diverse samples for ambiguous inputs. Performance benchmarks on the ShapeNet dataset report negative log-likelihoods of $2.46$–$2.50$ bits per vertex for vertices and $1.81$–$1.87$ for faces, with predictive accuracy near $85$% for vertex prediction and $90$% for face index prediction. PolyGen demonstrates competitive sample-based reconstruction performance (as measured by symmetric Chamfer distance) relative to surface optimization models such as AtlasNet.

Its applications span automated asset creation for graphics and games, robust object reconstruction in robotics, and efficient mesh representation for virtual worlds. By directly linking mesh geometry and topology, PolyGen obviates intermediate representations such as voxels or point clouds, enabling efficient, high-fidelity mesh synthesis.

2. polyGen: Generative Framework for Atomic-level Polymer Structures

“polyGen: A Learning Framework for Atomic-level Polymer Structure Generation” (Jain et al., 24 Apr 2025) presents the first generative model expressly designed for atomic-level polymer conformations from minimal chemistry input, such as the repeat unit.

polyGen’s framework encodes polymer connectivity as a graph ($G$), which is processed via a conditional encoder ($\mathcal{E}_c$) yielding positional and bonding encodings. Generation proceeds in latent space, where a diffusion model integrated within a Transformer (exploiting positional-biased attention) iteratively denoises random latent variables. Positional bias ensures attention mechanisms reflect local chemical and geometric relationships, preserving the underlying physical integrity required for realistic conformations.

The training protocol addresses limited dataset size (3,855 DFT-optimized structures) through joint learning with small molecule datasets, with loss terms assigned for bounding box, bond, angle, dihedral, and fractional coordinate reconstruction. The model employs a latent trajectory dictated by a discretized ODE/SDE, with “churn” noise added to avoid poor local minima: $$\begin{align*} \Delta t &= 1/T\ \Delta\tilde{t} &= \gamma(1 - t)\ \tilde{t} &= \max(t - \Delta\tilde{t}, 0)\ Z_t &\leftarrow Z_t + \Delta\tilde{t} \sqrt{t^2 - \tilde{t}^2}\varepsilon\ \hat{u}_t &= (\tilde{\mathcal{Z}}_1 - Z_t)/(1 - t)\ Z_{t+\Delta t} &= Z_t + \hat{u}_t \Delta t \end{align*}$$ where $Z_t$ denotes the latent, $\varepsilon$ is random noise, and $\gamma$ modulates noise intensity.

Structural matching metrics (incorporating synthetic accessibility and local geometric correctness) benchmarked polyGen against traditional polymer structure prediction algorithms. The framework generated realistic linear and branched conformations, even for large repeat units, outperforming rule-based and crystalline pathway methods.

3. PolyGen and Related Models in Polymer Informatics

PolyGen, as a generative approach for polymers, contrasts with hierarchical methods such as PolyConf (Wang et al., 11 Apr 2025). PolyConf decomposes polymer structure into repeating unit conformations (generated by a masked autoregressive model) and orientation transformations (assembled via an SO(3) diffusion model). While polyGen operates primarily on atomic-level representation, PolyConf advances hierarchical generative modeling, contributing a benchmark dataset (PolyBench) to facilitate standardized evaluation of polymer conformation generation.

4. Mathematical PolyGen: Degenerate Multi-poly-Genocchi Polynomials

PolyGen also denotes mathematical objects in combinatorics and number theory, most notably degenerate multi-poly-Genocchi polynomials (Kim et al., 2020, Corcino et al., 2020). These polynomials are defined via generating functions built from degenerate multiple polyexponential and polylogarithm functions: $$E_{k_1, \ldots, k_r, x}^{(\mathrm{deg})}(t) = \sum_{0 < n_1 < \cdots < n_r} \frac{(n_1-1)! \cdots (n_r-1)!}{n_1^{k_1}\cdots n_r^{k_r}} t^{n_r}$$ The degenerate multi-poly-Genocchi polynomials $g_n^{(k_1, \ldots, k_r)}(x)$ are characterized by explicit expansion formulas, convolution identities, and connections with degenerate Stirling numbers.

Generalizations include the Apostol-type poly-Genocchi polynomials with parameters $a, b, c$ (Corcino et al., 2020), which admit recursive and differential identities and form Appell polynomial families. Both ordinary and type-2 variants are explored, the latter via polyexponential generating functions, giving rise to further identities involving Stirling numbers and poly-Bernoulli analogs.

5. Applications and Implications

PolyGen’s Transformer-based mesh generator (Nash et al., 2020) advances state-of-the-art practice in asset creation, virtual environment generation, and robotic perception. The atomic-level polymer generative models (Jain et al., 24 Apr 2025) and hierarchical approaches (Wang et al., 11 Apr 2025) accelerate the development and screening of polymers, addressing challenges in molecular diversity, conformation flexibility, and rapid materials discovery.

Mathematical PolyGen polynomials underpin advances in combinatorial identities, special function theory, and computational mathematics. Their explicit formulae and convolution properties broaden the toolkit for researchers in analytic number theory and related fields.

6. Open Research Directions

PolyGen and its descendants inspire several open avenues:

• Scaling 3D generative models to higher mesh resolutions or complex surface topologies;
• Enhancing polymer informatics by integrating expanded datasets, better positional attention, and advanced autoregressive/diffusion architectures;
• Extending the mathematical frameworks for poly-Genocchi polynomials to multivariate, $q$-analog, or degenerate settings with broader applicability;
• Hybridizing robust learning frameworks with domain knowledge to further improve generalization and sample diversity.

The intersection of generative modeling and mathematical function theory embodied in PolyGen underscores its multidisciplinary importance and opens prospects for further theoretical and applied advancements.