AI-Kolmogorov: Neural Networks & Complexity

• AI-Kolmogorov is a research framework that integrates learnable neural architectures inspired by the Kolmogorov-Arnold theorem with principles from algorithmic complexity to produce compact, interpretable models.
• It leverages Kolmogorov-Arnold networks (KANs), replacing fixed scalars with adaptive univariate functions, thereby enhancing efficiency in data fitting, PDE solving, and equation discovery.
• The paradigm extends across diverse applications including hydrology, systems biology, optimization, and AI safety, emphasizing symbolic extraction and verifiability in model selection.

“AI-Kolmogorov” (Editor's term) denotes a family of research programs that connect artificial intelligence with two distinct mathematical lineages associated with A. N. Kolmogorov. One lineage is architectural: Kolmogorov-Arnold networks (KANs) are neural networks inspired by the Kolmogorov-Arnold representation theorem, replacing scalar weights and fixed node activations with learnable univariate functions on edges (Liu et al., 2024). The other lineage is informational: Kolmogorov complexity and algorithmic information theory formalize the information content of an individual object by the length of its shortest effective description, and use that formalism to study randomness, model selection, prediction, cryptographic asymmetry, and limits of verification (Shen, 2015, 0809.2754). In current work, these lineages appear in interpretable equation discovery, PDE solvers, hydrological constitutive-law replacement, signed gene-regulatory inference, perception trust surrogates, constraint-informed control, and theoretical analyses of AI safety certification (Liu et al., 2024, Wang et al., 2024, Tong et al., 16 Jun 2025, Hasan, 6 Apr 2026).

1. Dual lineage: representation and complexity

The KAN lineage begins from the observation that a multivariate function can be represented through sums and compositions of univariate functions. In the canonical KAN formulation, a layer is written as

$x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$

so the node performs summation while each edge carries a learnable univariate function. The original KAN paper states the architectural inversion explicitly: MLPs have fixed activations on nodes, whereas KANs have learnable activations on edges, and “have no linear weights at all — every weight parameter is replaced by a learnable univariate function parametrized as a spline” (Liu et al., 2024).

The complexity lineage uses shortest-description length as the primitive notion of information. In the standard formulations summarized in the survey literature,

$$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$

with $C$ the plain complexity and $K$ the prefix complexity for a fixed optimal universal machine $U$. This framework supports incompressibility arguments, randomness criteria, universal semimeasures, structure functions, and Chaitin-style incompleteness phenomena (Shen, 2015, 0809.2754).

These two lineages are mathematically distinct. One concerns neural parameterization of functions; the other concerns description length, randomness, and provability. Their current conjunction in AI research suggests a common emphasis on compact structure, interpretable decomposition, and explicit control over what counts as explanation rather than raw prediction.

2. Kolmogorov-Arnold networks as an interpretable neural paradigm

KANs operationalize the Kolmogorov-Arnold representation theorem by turning edge-wise univariate maps into trainable objects. The original construction parameterizes each edge function by a base function and a spline term,

$\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$

with the spline expanded in B-spline basis functions. The same paper argues that smaller KANs can achieve comparable or better accuracy than much larger MLPs on data fitting and PDE solving, that KANs possess faster neural scaling laws than MLPs, and that the learned edge functions are directly visualizable (Liu et al., 2024).

This architecture has already been extended in explicitly science-facing directions. “KAN 2.0” introduces MultKAN, which adds multiplication nodes; kanpiler, which compiles symbolic formulas into KANs; and a tree converter that converts KANs or any neural networks to tree graphs. The stated use cases are identifying relevant features, revealing modular structures, and discovering symbolic formulas, with demonstrations on conserved quantities, Lagrangians, symmetries, and constitutive laws (Liu et al., 2024).

A parallel line of work argues that standard KANs are inflexible for ridge functions. The asKAN paper studies this limitation through active subspaces and proposes active subspace embedded KAN, which inserts active-subspace detection between KAN blocks so that the model learns primary ridge directions and projects inputs onto critical dimensions without increasing the number of neurons in the original KAN (Zhou et al., 7 Apr 2025). This is a significant correction to overly broad claims of KAN universality in scientific settings: the architecture is expressive, but its default coordinate-wise decomposition can be mismatched to functions that depend mainly on low-dimensional linear combinations of inputs.

3. Equation discovery and AI-for-science

A prominent AI-Kolmogorov use case is symbolic or semi-symbolic scientific law refinement. In hydrology, KANs are used as an explainable AI / symbolic regression method to replace semi-empirical constitutive laws for baseflow identification. Using 378 CAMELS catchments across the contiguous United States and the aridity index

$\phi=\mathrm{PET}/P,$

the paper replaces the original normalized baseflow law by the KAN-derived symbolic relation

$$\frac{Q_B}{P}=0.39 - 0.34\, \text{tanh}(1.42\phi-0.82),$$

and then proposes the simplified domain-guided form

$f_B(\phi)=0.7573 - 0.7243\tanh(\phi).$

On the test set, the normalized KAN-derived relation improves NSE from 0.432 to 0.741, reduces RMSE from 0.0866 to 0.0585, improves KGE from 0.636 to 0.795, and changes $R^2$ from 0.733 to 0.744. For the refined absolute baseflow relation $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$0, the reported metrics improve from NSE 0.432 to 0.886, KGE 0.636 to 0.920, and $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$1 0.733 to 0.886. The same study emphasizes that the learned equations are closed-form, simpler than the original relation, and directly writable into hydrological models (Liu et al., 2024).

In AI for PDEs, Kolmogorov-Arnold-Informed Neural Networks (KINN) replace the MLP backbone in PINN-, DEM-, and BINN-style solvers with KANs. The framework is evaluated on multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. The paper reports that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. In the Mode III crack benchmark, the reported relative errors are 0.062535 for CPINNs_MLP versus 0.029879 for KINN-CPINNs, 0.024292 for DEM_MLP versus 0.011166 for KINN-DEM, and 0.000872 for BINN_MLP versus 0.000828 for KINN-BINN (Wang et al., 2024).

In systems biology, scKAN replaces tree predictors in One-vs-Rest gene regulatory network inference with differentiable KANs. The model is trained on BEELINE and uses gradients of the learned predictor to infer both edge strength and regulation sign: positive gradient for activation, negative gradient for inhibition. The paper reports that scKAN surpasses the leading signed GRN inference models by 5.40% to 28.37% in AUROC and 1.97% to 40.45% in AUPRC, while remaining scalable to larger CollecTRI-derived networks (Tong et al., 16 Jun 2025).

4. Perception, detection, and trust surrogates

In computer vision, KANs have been used both as primary classifiers and as post-hoc interpretable surrogates. For AI-generated image detection, one paper combines CLIP semantic image embeddings with either a plain MLP baseline or a hybrid KAN+MLP classifier. The training dataset contains 1000 real images from RAISE 1k, 340 Stable Diffusion 3 Ultra images, 333 DALL-E 3 images, and 333 MidJourney 6 images; the out-of-distribution test set contains 500 real images, 500 Adobe Firefly images, 500 DALL-E 3 images, and 500 MidJourney 5 images. The hybrid model achieves F1 scores of 0.94 for real versus DALL-E 3, 0.94 for real versus MidJourney 5, and 0.91 for real versus Adobe Firefly, compared with 0.93, 0.93, and 0.89 for the baseline MLP (Anon et al., 2024).

A different perception use treats KAN as an interpretable surrogate for detector confidence. In the YOLOv10 study, the KAN receives seven inputs derived from each detection: normalized $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$2, normalized $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$3, normalized width $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$4, normalized height $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$5, predicted confidence $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$6, discrete class index $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$7, and relative image scale $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$8. The surrogate has architecture $$C(x)=\min\{|p|:U(p)=x\}, \qquad K(x)=\min\{|q|:U(q)=x\},$$9, grid $C$0, and spline order $C$1, and is trained on 9098 detections. Its reported surrogate fidelity is

$C$2

The paper interprets lower fidelity in the lowest and highest confidence bins as evidence of edge-case regions where trust is less secure, and qualitatively highlights blur, partial occlusion, clutter, and low texture as low-trust regimes. BLIP captioning is added as a lightweight multimodal interface that does not alter the interpretability layer (Impraimakis et al., 24 Mar 2026).

KANs have also been tested in art authentication. In the Beltracchi forgery study, the task is not binary “authentic vs. fake” classification but 12-way multiclass classification over 11 forged artists plus Wolfgang Beltracchi. The final dataset contains 1334 paintings and 59672 patches. EfficientNet variants achieve around 77% validation accuracy, whereas tested KAN configurations reach about 45–53%. Even so, the paper reports substantial agreement between KAN and EfficientNet on suspicious works such as “Banks of Seine,” “London Bridge,” and “Still Life with Pears and Indian Bowl,” all attributed to André Derain in the dataset (Boccuzzo et al., 2024).

5. Optimization, control, and uncertainty-aware digital replicas

KANs have also been positioned as interpretable surrogates for optimization under uncertainty. In smart-grid research, a general KAN framework is proposed for problems of the form

$C$3

with uncertainty vector $C$4, decision variables $C$5, and state variables $C$6. The trained KAN learns the mapping

$C$7

thereby serving as a digital replica of the underlying optimization process. The case study is a stochastic optimal power flow problem in a modified IEEE 5-bus hybrid AC/DC system with Beta-distributed solar irradiation, Gaussian load uncertainty, and constraints including voltage limits, thermal limits, power balance, AC/DC conversion constraints, and branch flow limits. The paper reports that three-layer KAN networks provide accurate estimation, that learned activation functions capture statistical features and correlations of uncertain inputs, and that PDFs, CDFs, means, variances, and confidence intervals match Monte Carlo baselines well (Wang et al., 2024).

In autonomous spacecraft rendezvous, Constraint-Informed Kolmogorov-Arnold Networks (CIKAN) approximate the Time Shift Governor (TSG) map. The original TSG selects a time shift $C$8 so that a nominal controller tracks a time-shifted reference trajectory while satisfying state and input constraints. The learned surrogate predicts the optimal shift offline and is combined with a fallback constrained optimization step when the candidate shift is not admissible. In the reported highly elliptic orbit scenario, conventional TSG has average computation time 0.0871 s, worst-case computation time 0.8546 s, and average Delta-V 3.8741 km/s. The reported CIKAN $C$9 result has 1,638,400 parameters, train and validation loss $K$0, RMSE 0.9652 s, average computation time 0.0329 s, worst-case time 0.5590 s, and average Delta-V 3.4965 km/s. Among KAN variants, GRBF-CIKAN is reported to have the best validation loss and the smallest number of parameters (Kim et al., 2024).

6. Algorithmic information theory, model selection, and prediction

The second major AI-Kolmogorov lineage is algorithmic information theory. Its basic objects are individual strings rather than parametric models or statistical ensembles. The survey literature emphasizes that most strings are incompressible, that prefix complexity satisfies

$K$1

for the universal a priori semimeasure $K$2, and that an infinite sequence is Martin-Löf random exactly when there exists a constant $K$3 such that every prefix of length $K$4 has prefix complexity at least $K$5 (Shen, 2015).

This framework has been used to reinterpret statistical evidence and model selection. One synthesis states that small probability alone is not enough to reject a hypothesis; the observed event must also be simple enough to be meaningfully singled out. Its formulation is explicit: events that are both simple and very improbable should not happen, and if they do, the hypothesis is discredited. A companion survey develops this into algorithmic statistics through the structure function

$K$6

together with algorithmic sufficient statistics and the MDL-style criterion $K$7, presented as a formalization of Occam’s razor in inductive inference (Shen, 2009, 0809.2754).

The same description-length viewpoint also yields non-probabilistic predictive theory. Universal probability-free prediction defines laws of nature as recursively enumerable prefix-free subsets of finite data sequences and constructs universal prediction systems that dominate any fixed prediction system up to constant factors. The same work develops time complexity, time randomness deficiency, and universal randomness-type prediction systems, and states that under the IID assumption these probability-free predictors dominate conformal prediction to within the usual accuracy (Vovk et al., 2016).

Conditional complexity has also been used to formalize asymmetric information release. One cryptographic result shows that, for strings $K$8 and $K$9, there exists a message $U$0 such that $U$1 is reconstructible from $U$2 while $U$3 alone reveals as little about $U$4 as information-theoretically possible; the paper then generalizes this to adversaries with auxiliary side information $U$5 and derives conditions under which a polynomial-size message exists (Muchnik, 2011).

7. Limits, caveats, and theoretical boundaries

The sharpest theoretical boundary in this literature is a Kolmogorov-complexity-based incompleteness theorem for AI safety verification. The formalization encodes an AI interaction as

$U$6

where $U$7 is the input, $U$8 the system output, and $U$9 the policy specification, and defines policy compliance by

$\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$0

For a fixed sound computably enumerable theory $\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$1, the main theorem states that there exists a constant $\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$2 such that for all sufficiently large $\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$3, no statement of the form

$\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$4

is provable in $\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$5. The paper stresses that this is not a computational hardness result: the barrier is information-theoretic rather than resource-theoretic. Its proposed workaround is proof-carrying, instance-level certification, in which the encoding is augmented to $\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$6 with an explicit proof $\phi(x)= w_b\, b(x) + w_s\, \mathrm{spline}(x),$7 that can be checked even though no fixed verifier can be complete over arbitrarily high-complexity compliant instances (Hasan, 6 Apr 2026).

Empirical KAN research also has clear limitations. The asKAN paper identifies standard KAN inflexibility for ridge functions (Zhou et al., 7 Apr 2025). KINN reports an explicit exception on the complex geometry problem (Wang et al., 2024). In art authentication, the tested KAN models are markedly worse than EfficientNet in validation accuracy (Boccuzzo et al., 2024). In GRN inference, scKAN is evaluated only on synthetic and curated BEELINE data, cannot detect self-loops because each target gene is removed from its own predictor, and remains weak on BF, BFC, and TF datasets (Tong et al., 16 Jun 2025). In spacecraft control, approximation error is acknowledged as a threat to recursive feasibility and constraint satisfaction, and formal robustness guarantees are left open (Kim et al., 2024). These results suggest that AI-Kolmogorov methods are strongest when the target relation has exploitable low-dimensional or compositional structure and when interpretability or symbolic extraction is a first-class objective, rather than a secondary diagnostic.