COLE: Diverse Mathematical & Computational Models

• COLE is a multifaceted term encompassing classical constructs like the Cole–Hopf transform and Cole–Cole law, as well as modern architectures for design, neural search, and blockchain.
• The mathematical lineage of COLE illustrates how logarithmic-derivative methods linearize nonlinear dynamics and model fractional relaxation in complex systems.
• Modern applications of COLE demonstrate significant practical benefits including reduced evaluation budgets in neural architecture search, enhanced graphic design pipelines, and efficient blockchain storage.

In arXiv literature, COLE does not denote a single object. It names a family of mathematically and technically distinct constructs: the Cole–Hopf mechanism relating linear diffusion to nonlinear Burgers dynamics, the Cole–Cole law of anomalous relaxation and dispersive media, and several modern systems and benchmarks, including a modular architecture for automatic graphic design, a code-based embedding method for surrogate-assisted neural architecture search, a column-based learned blockchain storage architecture, and a comprehensive benchmark for French natural language understanding (Iwata, 2018, Górska et al., 2018, Inoue et al., 2024, Somu et al., 15 May 2026, Zhang et al., 28 Feb 2026, Beauchemin et al., 6 Oct 2025).

1. Mathematical lineage: the Cole–Hopf transform and its generalizations

The classical Cole–Hopf transform is the prototype from which many later “COLE” usages derive. In the one-dimensional setting reviewed in the abstract formulation paper, if $u(t,x)$ solves the heat equation

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$

then

$\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$

solves the Burgers equation

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$

The paper reinterprets this not as an isolated substitution but as a logarithmic-derivative mechanism: nonlinearity emerges from expressions of the form $u_x/u$. Its main operator-theoretic contribution is an abstract representation of infinitesimal generators of invertible evolution families in Banach spaces, together with a “relativistic formulation” in which the evolution direction may be any coordinate $x^i$, not only time. In that setting, the generalized Cole–Hopf transform is expressed through

$$A(x^i)U(x^i,\xi^i)u_{\xi^i}=(KI+U(x^i,\xi^i))\,[\partial_{x^i}\log(U(x^i,\xi^i)+KI)]u_{\xi^i},$$

thereby identifying the classical transform as a special case of a logarithmic representation of generators (Iwata, 2018).

Several later papers extend the same principle to wider nonlinear classes. One line studies variable-coefficient Burgers-type PDEs and second-order nonlinear ODEs with convective terms via affine logarithmic-derivative ansätze such as

$$\psi=P(x)+Q(x)\frac{\phi_x}{\phi},\qquad \psi=P(x)+Q(x)\frac{\phi'}{\phi},$$

and derives explicit coefficient constraints under which nonlinear equations collapse to linear diffusion or linear second-order ODEs (Humi, 2013). A closely related “hybrid Cole–Hopf–Darboux transformation” connects nonlinear second-order ODEs directly to linear second-order ODEs of the same order through

$\psi(x)=P(x)+Q(x)\frac{\phi'(x)}{\phi(x)},$

with applications to equations associated with Airy, Bessel, Hermite, Legendre, and Laguerre functions, and to a special Painlevé II case with $a=-\tfrac12$ (Humi, 2012). A further generalization applies the same affine-logarithmic template to perturbed Van der Pol and Liénard equations, a restricted Painlevé III family, and generalized Burgers and convective equations, again treating linearizability as a coefficient-matching problem rather than a universal property of the full nonlinear class (Humi, 2014).

2. Contemporary Hopf–Cole applications beyond classical Burgers theory

The Cole–Hopf mechanism remains active in contemporary applied analysis, but usually in specialized rather than universal forms. In one nonlinear Schrödinger-type study, the equation

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$0

is rewritten as

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$1

which is precisely a complex Burgers equation. The standard Cole–Hopf substitution

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$2

then reduces it to

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$3

A common misconception is that this solves the standard cubic NLS; the paper instead treats only a special derivative-nonlinearity equation of Burgers type (Rezazadeh et al., 2010).

In kinetic neural-network theory, a Hopf–Cole transform is used at a different scale. For a spatially extended FitzHugh–Nagumo model in the strong-interaction regime, the transform

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$4

exposes the concentration exponent and yields local-uniform estimates proving that the blow-up profile in the voltage variable is Gaussian. Under well-prepared initial data, the paper establishes

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$5

so that $\partial_t u-\mu^{1/2}\partial_x^2u=0,$6 concentrates on $\partial_t u-\mu^{1/2}\partial_x^2u=0,$7 with an explicit Gaussian profile rather than merely in weak or integral senses (Blaustein et al., 2022).

Two more recent computational works embed Hopf–Cole-type ideas into numerical schemes. A fourth-order multiple-relaxation-time lattice Boltzmann model for the $\partial_t u-\mu^{1/2}\partial_x^2u=0,$8-dimensional coupled Burgers equations first transforms

$\partial_t u-\mu^{1/2}\partial_x^2u=0,$9

so that $\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$0 satisfies a diffusion equation, then designs a D$\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$1Q$\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$2 MRT-LB model whose modified equation is fourth order and whose non-equilibrium distribution yields $\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$3 locally with fourth-order accuracy; the reported simulations in $\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$4 show a fourth-order convergence rate (Chen et al., 2023). In porous-media gas flow, a machine-learning-enhanced formulation replaces pressure by

$\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$5

which exactly absorbs the Klinkenberg factor $\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$6 and converts the original nonlinear mixed flow equations into a linear Darcy-type system in $\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$7; physical pressure is then recovered through a Lambert-$\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$8 inversion (Maduru et al., 11 Mar 2026).

3. Cole–Cole relaxation, fractional evolution, and dispersive electromagnetics

A second major mathematical lineage is Cole–Cole, not Cole–Hopf. In the relaxation setting, the defining time-domain law is

$\psi(t,x)=-2\mu^{-1/2}\partial_x\log u(t,x)$9

with Debye relaxation recovered at $$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$0. The central structural result is that Cole–Cole relaxation still obeys a composition principle, but not by ordinary multiplication. The correct law is

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$1

where $$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$2 is realized by an integro-differential operation rather than a semigroup product. The same structure is shown to be equivalent to the fractional evolution equations

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$3

thereby identifying Cole–Cole relaxation as a non-Markovian fractional kinetics law rather than a multiplicative memoryless semigroup (Górska et al., 2018).

In computational electromagnetics, this constitutive law appears as a fractional-memory polarization equation. One discontinuous Galerkin study considers the one-dimensional time-domain Maxwell system with

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$4

introduces a diffusive representation of the Caputo kernel, and proves that the physically relevant energy is

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$5

with $$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$6 accounting for the continuum of internal relaxation modes. The fractional convolution is then approximated by positive quadrature coefficients $$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$7, chosen by nonlinear constrained optimization so that the approximate system remains passive and energy-decaying under a DG spatial discretization and BDF2 time stepping (Xie et al., 2022).

A later analysis for the two-dimensional Maxwell system in a bounded rectangular domain sharpens the energy-decay statement further. It proves the continuous law

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$8

for a modified energy containing the history term

$$\partial_t\psi+\psi\,\partial_x\psi=\mu^{-1/2}\partial_x^2\psi.$$9

and then constructs a shifted fractional trapezoidal rule, the SFTR-$u_x/u$0 scheme, with a discrete energy dissipation theorem valid for

$u_x/u$1

The temporal convergence rate is first order for $u_x/u$2 and second order for $u_x/u$3, and the comparison with a second-order fractional backward difference formula shows markedly better long-time monotone energy decay (Zhang et al., 11 Dec 2025).

4. COLE as a modular architecture for automatic graphic design

In machine learning for visual communication, COLE denotes a staged system for automatic graphic design generation from short user intentions. As described by the open reimplementation paper, the task is not generic text-to-image synthesis but production of a complete design artifact—poster, advertisement, cover, or social-media graphic—through a layered pipeline that separates semantic planning, non-text imagery, typography, and rendering. COLE decomposes generation into four stages: interpretation of the user intention into a structured design plan, generation of the visual imagery, generation of typography attributes for editable text layers, and final rendering. The output is conceptually layered into a background layer, an object-image layer, and text layers with typographic properties, so that text remains legible and editable rather than being rasterized into the image (Inoue et al., 2024).

OpenCOLE reproduces this design philosophy using public resources only. It replaces COLE’s fine-tuned Llama plan generator with GPT3.5 in-context learning using 5 user-intention/design-plan pairs; extracts design-plan fields from images with LLaVA-1.5-13B by a divide-and-conquer prompting strategy; replaces COLE’s two-stage background/object image generation with a single-stage SDXL 1.0 fine-tuning; and fine-tunes LLaVA1.5-7B for typography generation. Its base dataset is Crello, described as around 22k vector-format design templates containing images, texts, layouts, and typography information. On the DESIGNERINTENTION benchmark of 200 prompts, evaluated by GPT4V on design and layout, content relevance, typography and color, graphics and images, and innovation, the reported averages are 6.0 for COLE and 6.3 for OpenCOLE. The paper also emphasizes persistent limitations: dependence on a black-box GPT4V assessor, low legibility text, overly long sentences without line breaks, poor color contrast, and thin fonts (Inoue et al., 2024).

5. COLE as representation and storage infrastructure

A very different modern usage is COLE as Code-Oriented LM Embeddings for surrogate-assisted neural architecture search. In this formulation, an architecture is deterministically converted into a PyTorch class definition, passed through a frozen LLM, embedded by mean pooling of last-layer hidden states,

$u_x/u$4

reduced by PCA to 128 dimensions, and scored by a 3-layer MLP. The paper evaluates this representation on NAS-Bench-201 and einspace, and uses it as a drop-in replacement for BANANAS’s path encoding. Its headline downstream result is specific: on NAS-Bench-201 for CIFAR-100 test accuracy, replacing path encodings with COLE reduces the number of true architecture evaluations needed to get within 1% of the best architecture in the search space, from 200 to 132, a 34% reduction in evaluation budget (Somu et al., 15 May 2026).

In blockchain systems, COLE names a column-based learned storage architecture for authenticated historical state. The design uses an LSM-tree update engine with compound keys

$u_x/u$5

an in-memory Merkle B-tree, and on-disk runs composed of a value file, an index file containing $u_x/u$6-bounded piecewise linear models with

$u_x/u$7

and a Merkle file authenticating the run. The motivation is to avoid Merkle Patricia Trie duplication across state versions. The later COLE$u_x/u$8 paper treats this original COLE as a strong but incomplete starting point, adding a rewind-supported in-memory RS-tree based on content-defined chunking, a two-level Merkle Hash Tree structure, and a prunable version tree so that chain reorganization and state pruning become practical. The paper reports that original COLE had already achieved up to $u_x/u$9 storage reduction over MPT, while COLE$x^i$0 reaches up to $x^i$1 smaller storage than COLE and $x^i$2 smaller than MPT (Zhang et al., 28 Feb 2026).

6. COLE as a French NLU benchmark, and related names often confused with it

In language evaluation, COLE stands for COrpus for Langue understanding Evaluation. It is a French NLU benchmark comprising 23 tasks grouped into single-sentence tasks, similarity and paraphrase tasks, and inference tasks, and the paper evaluates 95 LLMs with a benchmark-wide Composite Score defined as the unweighted mean of per-task scores. The task set spans sentiment analysis, paraphrase detection, semantic textual similarity, grammatical acceptability, natural language inference, extractive and boolean question answering, pronoun resolution, word-sense disambiguation, and Quebec French definition-matching and grammaticality tasks. The best reported Composite Score is 70.12 for GPT-5-mini-2025-08-07; the random baseline is 31.22; and the best open-weight model reported is Qwen-max at 49.14. The benchmark’s diagnostic result is not merely ranking: it identifies zero-shot extractive QA, fine-grained WSD, and regional language variation as especially difficult frontiers for current models (Beauchemin et al., 6 Oct 2025).

Several adjacent names are distinct and should not be conflated with COLE proper. CoLES, for example, is Contrastive Learning for Event Sequences, a self-supervised representation-learning method in which random contiguous subsequences from the same event history are treated as positive pairs and trained with a classical margin-based contrastive loss; it is orthographically similar but methodologically unrelated to the other COLE usages (Babaev et al., 2020). AL-CoLe is Augmented Lagrangian for Constrained Learning, a constrained-learning framework based on an augmented Lagrangian

$x^i$3

introduced for non-convex statistical constrained learning and fairness-constrained classification; despite the similar typography, it belongs to a different optimization lineage (Boero et al., 23 Oct 2025).

Across these usages, the term therefore functions less as a single concept than as a recurrent label attached to several influential structures: a logarithmic linearization mechanism in nonlinear analysis, a fractional relaxation law in dielectric media, and a set of modern architectures, embeddings, storage engines, and evaluation suites in machine learning and computer systems.