Kolmogorov-Arnold Additive Model (KAAM)

• KAAM is a structured additive-composition model where a multivariate function is represented as a finite sum of univariate outer functions applied to sums of univariate inner functions.
• Operationalized via Kolmogorov-Arnold Networks (KANs), KAAM replaces scalar weights with learnable univariate functions, typically parameterized by splines, to capture multivariate dependencies.
• KAAM offers strong theoretical guarantees including dimension-independent error rates and has diverse applications in regression, dynamical systems, and clinical classification.

Searching arXiv for papers on Kolmogorov–Arnold additive models and networks to ground the article in current literature. Kolmogorov–Arnold Additive Model (KAAM) denotes a family of multivariate function models derived from the Kolmogorov–Arnold representation theorem, according to which a continuous function on a bounded domain can be expressed as a finite sum of univariate outer functions applied to sums of univariate inner functions. In its canonical form, KAAM is neither a purely coordinate-wise additive model nor a conventional multilayer perceptron: it is an additive-composition model in which multivariate dependence is mediated by nested sums and univariate compositions. In contemporary machine learning, KAAM is most commonly operationalized through Kolmogorov–Arnold Networks (KANs), which replace scalar weights by learnable univariate functions, usually parameterized by splines (Liu et al., 2024). Subsequent literature has broadened the term’s scope: some works retain the canonical nested form for nonparametric regression and scientific model discovery, while others use “KAAM” more narrowly for explicitly additive, single-layer KAN variants designed for interpretability in domains such as clinical classification (Liu et al., 24 Sep 2025, Almodóvar et al., 20 Sep 2025).

1. Formal definition and terminological scope

The canonical Kolmogorov–Arnold representation used in modern KAAM literature is

$$f(x_1,\dots,x_n)=\sum_{q=1}^{2n+1}\Phi_q\!\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right),$$

with $f:[0,1]^n\to\mathbb{R}$ continuous on a bounded domain and all $\phi_{q,p}$ and $\Phi_q$ continuous univariate functions. The theorem fixes the outer width at $2n+1$ and yields what is effectively a depth-2 nonlinear architecture: univariate inner maps, an inner sum, an outer univariate map, and a final outer sum (Liu et al., 2024). A recurring caveat is that the theorem is existential rather than constructive: the guaranteed univariate functions can be highly non-smooth, even fractal, which complicates learnability in practical settings (Liu et al., 2024).

KAAM differs materially from generalized additive models. A GAM typically uses

$f(x)=\sum_{i=1}^{n} g_i(x_i),$

whereas canonical KAAM inserts an additional compositional stage,

$$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$

with $Q=2n+1$ in the classical theorem. The model is therefore additive inside a composition, not merely additive across coordinates (Liu et al., 2024). This distinction is central in later statistical analyses, which treat the univariate components as the estimands that make the multivariate problem tractable (Liu et al., 24 Sep 2025).

The term “KAAM” is not fully uniform across the literature. In some work it denotes the theorem-faithful nested additive-composition class; in other work it denotes a constrained additive KAN with no outer compositional stage beyond the logistic link or final sum. The following formulations all appear in current usage.

Context	Formulation	Emphasis
Canonical theorem/KAN usage	$\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$	Additive-composition representation
Nonparametric regression usage	$\sum_q g_q(\sum_j \psi_{qj}(x_j))$	Sobolev regularity and minimax rates
Clinical classification usage	$f:[0,1]^n\to\mathbb{R}$0	Single-layer additive interpretability

The older practical construction literature predating KANs already treated the Kolmogorov–Arnold representation as an explicit model class. One such formulation rewrote it as a tree of discrete Urysohn operators,

$f:[0,1]^n\to\mathbb{R}$1

and parameterized all univariate components as piecewise-linear functions (Polar et al., 2020). This confirms that KAAM is best understood as a modeling principle with multiple computational realizations rather than as a single architecture.

2. Neural operationalization: from KAAM to KANs

KANs make KAAM trainable by moving nonlinearity from nodes to edges. A KAN layer between layer $f:[0,1]^n\to\mathbb{R}$2 and $f:[0,1]^n\to\mathbb{R}$3 is a collection of learnable univariate edge functions $f:[0,1]^n\to\mathbb{R}$4, with forward propagation

$f:[0,1]^n\to\mathbb{R}$5

Composing layers yields

$f:[0,1]^n\to\mathbb{R}$6

A depth-2 KAN with shape $f:[0,1]^n\to\mathbb{R}$7 directly realizes the classical Kolmogorov–Arnold form, while deeper KANs generalize it through repeated additive-composition blocks (Liu et al., 2024).

In the original KAN construction, each edge function is parameterized as a spline,

$f:[0,1]^n\to\mathbb{R}$8

and, for optimization stability, is embedded in a residual activation,

$f:[0,1]^n\to\mathbb{R}$9

B-spline bases have local support, which supports both local refinement and direct visualization of learned edge functions. Initialization sets $\phi_{q,p}$0, initializes spline coefficients with small Gaussian noise, and uses Xavier-like schemes for $\phi_{q,p}$1; spline grids are updated during training so that preactivations do not drift outside the spline support (Liu et al., 2024).

This architecture gives KAAM a direct layer-wise semantics. A minimal two-stage representation is

$\phi_{q,p}$2

where each $\phi_{q,p}$3 is a learnable univariate spline. In deeper KANs, the outer $\phi_{q,p}$4 can itself be realized by another KAN layer, preserving the univariate-on-edge, sum-on-node paradigm (Liu et al., 2024).

A historically distinct operationalization is the Urysohn-tree construction, in which the inner functions $\phi_{q,p}$5 and outer functions $\phi_{q,p}$6 are piecewise-linear and updated by a record-by-record descent rule. Inputs are rescaled to interpolation segments, nodal values are updated by residual-driven corrections, auxiliary variables $\phi_{q,p}$7 are adjusted through derivative-aware steps, and the method extends to continuous, quantized, and mixed inputs (Polar et al., 2020). That formulation did not use the KAN terminology, but it implemented the same additive-composition structure.

The KAN paper emphasized that this seemingly simple change in parameterization alters both accuracy and interpretability. It reported that much smaller KANs can match or exceed much larger MLPs in data fitting and PDE solving, and that edge functions are visually inspectable in a way standard linear weights are not (Liu et al., 2024).

3. Approximation theory, statistical convergence, and scaling laws

Two distinct theoretical traditions now coexist around KAAM. One concerns approximation theory for spline-parameterized KANs; the other concerns statistical convergence of KAN regression estimators under Sobolev assumptions.

For spline KANs, a core approximation theorem assumes a representation

$\phi_{q,p}$8

with each $\phi_{q,p}$9 $\Phi_q$0-times continuously differentiable. There then exist $\Phi_q$1-th order B-spline approximations $\Phi_q$2 on grid size $\Phi_q$3 such that, for $\Phi_q$4,

$\Phi_q$5

For uniform approximation, the error scales as $\Phi_q$6 and is independent of ambient dimension $\Phi_q$7, provided a smooth KA representation exists (Liu et al., 2024). This dimension-independence is one reason the literature repeatedly states that KAAM can mitigate the curse of dimensionality when the target function truly admits a structured representation.

The same paper links this approximation behavior to neural scaling laws. If test RMSE scales as $\Phi_q$8 with parameter count $\Phi_q$9, the analysis yields $2n+1$0 for KANs under smooth KA representations, giving $2n+1$1 for cubic splines. Empirically, several synthetic tasks were reported to approach $2n+1$2, whereas MLPs plateaued earlier and scaled more slowly (Liu et al., 2024). A practical mechanism for reaching this regime is grid extension: during training, a coarse spline is fitted by a finer spline via least squares, producing staircase-like loss curves with sharp drops at each refinement stage (Liu et al., 2024).

The statistical literature formulates KAAM explicitly as a nonparametric regression class. In the setting

$2n+1$3

with $2n+1$4 i.i.d., mean zero, and finite variance, the additive KAN class is written

$2n+1$5

When the univariate components are bounded, Lipschitz, and represented by B-splines of order at least $2n+1$6, the estimator

$2n+1$7

achieves

$2n+1$8

which matches the one-dimensional minimax-optimal rate and is independent of $2n+1$9 (Liu et al., 24 Sep 2025). The same paper derives the optimal knot scaling

$f(x)=\sum_{i=1}^{n} g_i(x_i),$0

and extends the theory to hybrid additive–multiplicative nodes, where the rate remains $f(x)=\sum_{i=1}^{n} g_i(x_i),$1 up to a constant factor $f(x)=\sum_{i=1}^{n} g_i(x_i),$2 (Liu et al., 24 Sep 2025).

A related but architecturally distinct line of work introduces additive basis-expanded networks and hybrids with dense layers. In that framework, a KAAM is written

$f(x)=\sum_{i=1}^{n} g_i(x_i),$3

and instantiated by basis expansions with fixed Lipschitz activations mapping to $f(x)=\sum_{i=1}^{n} g_i(x_i),$4. Universal approximation results are proved for the shallow additive module and for hybrids that place additive blocks at the input, the output, or both (Kim et al., 2024). This suggests that the theorem-backed additive-composition principle is not tied exclusively to spline KANs.

4. Interpretability, symbolic structure, and applied scientific use

Interpretability is one of the principal reasons KAAM has attracted attention. Because the learned objects are univariate functions rather than opaque high-dimensional weight matrices, the model structure is inspectable at the level of individual edges, coordinates, and additive channels. In the original KAN study, edge magnitudes were used to identify important pathways, and pruning plus symbolic replacement produced compact formulas for multiplication, division, special-function compositions, and physics-inspired toy problems (Liu et al., 2024).

Two canonical examples illustrate the mechanism. A pruned $f(x)=\sum_{i=1}^{n} g_i(x_i),$5 KAN computes multiplication $f(x)=\sum_{i=1}^{n} g_i(x_i),$6 through quadratic and linear functions encoding

$f(x)=\sum_{i=1}^{n} g_i(x_i),$7

and a pruned $f(x)=\sum_{i=1}^{n} g_i(x_i),$8 KAN realizes positive division through

$f(x)=\sum_{i=1}^{n} g_i(x_i),$9

The same work described human-in-the-loop procedures built around sparsification, pruning, symbolification, and interactive testing of candidate edge functions such as linear, quadratic, $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$0, and $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$1, with discovery examples in knot theory and Anderson localization (Liu et al., 2024).

KAAM has also been used for scientific model discovery in dynamical systems. In that setting, KANs are trained as one-step predictors for discrete or continuous-time dynamics and then evaluated not only by rollout error but by invariant statistics such as Lyapunov exponents and Kullback–Leibler divergence between invariant measures. On the Ikeda map, two distinct KAN architectures reproduced the chaotic attractor with Lyapunov spectra close to the true values: true $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$2, learned $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$3 for a $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$4 model, and $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$5 for a $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$6 model (Moradi et al., 2024). That work emphasized non-uniqueness: multiple KAAM/KAN parameterizations can generate statistically equivalent dynamics, a point linked to shadowing in chaotic systems (Moradi et al., 2024).

In clinical tabular learning, “KAAM” is used more narrowly for a single-layer additive classifier. For class $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$7,

$$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$8

with probabilities obtained by the Logistic-KAN softmax or, in the binary case, the sigmoid of the differential logit (Almodóvar et al., 20 Sep 2025). This formulation sacrifices interaction modeling and universal approximation, but it yields direct patient-level decomposition through a class-specific logit matrix $$f(x_1,\dots,x_n)=\sum_{q=1}^{Q}\Phi_q\!\Big(\sum_{p=1}^{n}\phi_{q,p}(x_p)\Big),$$9 and, for binary classification, a differential logit matrix $Q=2n+1$0 (Almodóvar et al., 20 Sep 2025). The same study used partial dependence plots of $Q=2n+1$1, feature importance from column variances of $Q=2n+1$2, probability radar plots relative to an “average patient,” and nearest-patient retrieval via Euclidean distance in logit space (Almodóvar et al., 20 Sep 2025).

The clinical results were intended as an interpretability-performance trade-off rather than as a theorem-faithful reconstruction of the canonical KAAM. Across six health datasets, KAAM achieved competitive metrics, with overall mean reciprocal rank $Q=2n+1$3, best precision rank $Q=2n+1$4, and competitive ROC-AUC rank $Q=2n+1$5, while Logistic-KAN reached $Q=2n+1$6 overall (Almodóvar et al., 20 Sep 2025). A concrete Heart-dataset formula was reported in symbolic form, mixing linear terms for binary covariates with sinusoidal, cosine, and $Q=2n+1$7 terms for continuous features (Almodóvar et al., 20 Sep 2025).

5. Variants, extensions, and domain-specific generalizations

The KAAM principle has diversified rapidly. One axis of variation concerns the function space used for the univariate components. Activation space Selectable KAN (S-KAN) replaces the fixed spline basis by a selectable pool including B-splines, RBFs, orthogonal polynomials, wavelets, ReLU-like variants, and faster KAN variants. Training proceeds in three stages—full training, selective training of mixture weights, and pruning to a single activation family per node—and the selected model is then retrained from scratch (Yang et al., 2024). On seven function-fitting tasks, S-KAN either matched or improved the best fixed-activation baseline; on four image-classification datasets, S-ConvKAN achieved $Q=2n+1$8 on MNIST, $Q=2n+1$9 on Fashion-MNIST, $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$0 on CIFAR-10, and $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$1 on CIFAR-100 (Yang et al., 2024).

A second axis concerns adaptive model complexity. InfinityKAN treats the number of basis functions per univariate edge as a latent variable with a Poisson prior and learns it by variational inference. Each univariate function is expanded over a potentially unbounded basis family, and a differentiable weighting function gates the active basis count; the resulting ELBO is stated to be Lipschitz continuous with respect to changes in the number of active bases (Alesiani et al., 3 Jul 2025). Reported test accuracies included $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$2 on MNIST, $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$3 on FashionMNIST, $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$4 on CIFAR10, and $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$5 on EuroSAT, with performance often competitive with fixed-base KAN and generally above matched MLP baselines (Alesiani et al., 3 Jul 2025).

A third axis adds probabilistic structure. Sparse Variational Gaussian Process KANs place independent GP priors on univariate edge functions and propagate uncertainty through deep additive architectures by analytic moment matching for RBF kernels. Predictive mean and variance remain available at the level of each edge function, and the framework explicitly separates epistemic uncertainty from heteroscedastic aleatoric noise when a noise GP is added (Ju, 4 Dec 2025). In a fluid-flow calibration study, Pearson correlation between predicted uncertainty and absolute error was $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$6, coverage errors were below $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$7 percentage point across multiple confidence levels, the calibration error at $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$8 was $\sum_q \Phi_q(\sum_p \phi_{q,p}(x_p))$9 percentage points, and RMSE was $\sum_q g_q(\sum_j \psi_{qj}(x_j))$0 (Ju, 4 Dec 2025).

Other extensions embed KAAM into broader system designs. Geometry-aware R-Structured KANs add an analytic geometric gate $\sum_q g_q(\sum_j \psi_{qj}(x_j))$1 to a learned smooth KAN branch, yielding an explicitly additive model

$\sum_q g_q(\sum_j \psi_{qj}(x_j))$2

On an additive discontinuity benchmark over a rectangle, the geometry-aware additive model at $\sum_q g_q(\sum_j \psi_{qj}(x_j))$3 reduced test RMSE from $\sum_q g_q(\sum_j \psi_{qj}(x_j))$4 for standard KAN to $\sum_q g_q(\sum_j \psi_{qj}(x_j))$5, a $\sum_q g_q(\sum_j \psi_{qj}(x_j))$6 reduction, and boundary RMSE from $\sum_q g_q(\sum_j \psi_{qj}(x_j))$7 to $\sum_q g_q(\sum_j \psi_{qj}(x_j))$8 (Kucherenko et al., 1 Jul 2026).

In high-resolution vision, the Additive Kolmogorov–Arnold Transformer (AKT) replaces MLP blocks by Padé KAN modules and standard self-attention by PKAN Additive Attention. On the Point-based Maize Localization dataset of $\sum_q g_q(\sum_j \psi_{qj}(x_j))$9 UAV images with approximately $f:[0,1]^n\to\mathbb{R}$00 point annotations, AKT achieved an average F1-score of $f:[0,1]^n\to\mathbb{R}$01, outperforming state-of-the-art methods by $f:[0,1]^n\to\mathbb{R}$02, while reducing FLOPs by $f:[0,1]^n\to\mathbb{R}$03 and improving throughput by $f:[0,1]^n\to\mathbb{R}$04 (Li et al., 12 Jan 2026).

In dynamical forecasting, Kolmogorov–Arnold Reservoir Computing replaces recurrent reservoirs by explicit KA-inspired basis-function expansions over delay coordinates and trains only a linear readout by ridge regression. The first-order model is

$f:[0,1]^n\to\mathbb{R}$05

with higher-order versions adding polynomial outer interactions (Huang et al., 18 Jun 2026). On the double-scroll chaotic circuit, KARC achieved NRMSE $f:[0,1]^n\to\mathbb{R}$06 over the first Lyapunov time and a threshold time of $f:[0,1]^n\to\mathbb{R}$07 Lyapunov times, exceeding the corresponding values for RC and NG-RC (Huang et al., 18 Jun 2026).

There are also probabilistic generative reinterpretations. The Thermodynamic Kolmogorov–Arnold Model reinterprets the inner functions of the theorem as inverse CDFs of learned univariate energy-based priors, so that uniform inputs are pushed forward by deterministic Markov kernels into latent variables and then combined by a KAN generator. Training uses maximum likelihood, importance sampling or Langevin Monte Carlo for posterior expectations, and optionally thermodynamic integration or steppingstone estimation over power posteriors (Raj, 17 Jun 2025). This line of work treats the canonical KAAM structure as a generator design principle rather than a regression architecture.

Finally, kernel-based reinterpretations have connected KAAM to other model classes. One recent framework argues that KAN B-spline expansions, kernel methods, and self-attention can all be viewed as linear combinations of kernel evaluations, with self-attention corresponding to a linear-kernel special case. On CIFAR-10 under an MAE setup, the resulting Pseudo-MHSA retained comparable performance to a standard ViT of the same dimensionality while reducing parameters from $f:[0,1]^n\to\mathbb{R}$08M to $f:[0,1]^n\to\mathbb{R}$09M in the Param-Fusion variant and to $f:[0,1]^n\to\mathbb{R}$10M in the Semi-Fusion variant (Liu et al., 29 Mar 2025). This suggests that KAAM’s additive superposition can be recast in kernel language, though the paper frames this as a unifying interpretation rather than as an exact equivalence.

6. Limitations, misconceptions, and open problems

Several limitations recur across the literature. The strongest theoretical claims require smooth univariate components in the relevant KA representation. Yet the original theorem does not guarantee smooth inner functions, and the canonical representation may involve highly irregular functions that are difficult to approximate or optimize (Liu et al., 2024). A plausible implication is that practical success depends less on the theorem in its most general form than on whether the target task admits a smooth or at least learnable KA decomposition.

A second limitation is computational. Spline-based KANs were reported to be typically about $f:[0,1]^n\to\mathbb{R}$11 slower than MLPs with comparable parameter counts, largely because heterogeneous edge functions reduce batch efficiency (Liu et al., 2024). Grid refinement, adaptive knot placement, and richer basis families improve expressivity but also introduce optimization and overfitting issues, particularly near domain boundaries (Liu et al., 2024, Liu et al., 24 Sep 2025). The statistical convergence theory likewise depends on correct structural specification: if the target function does not admit a KAN-friendly additive-composition structure with Sobolev univariate components, the stated minimax rate no longer follows (Liu et al., 24 Sep 2025).

A common misconception is to identify KAAM with any additive model. In the canonical theorem-based sense, KAAM is not equivalent to a GAM, because it includes inner additive aggregation followed by an outer univariate composition. Conversely, some application papers deliberately restrict KAAM to a GAM-like single-layer additive form for interpretability, as in clinical classification (Almodóvar et al., 20 Sep 2025). Both uses are current, but they are not mathematically interchangeable.

Another misconception is that good one-step prediction implies unique mechanistic discovery. In chaotic model discovery, distinct KAN parameterizations produced invariant sets with essentially matching Lyapunov spectra and other statistics, showing that statistical equivalence does not imply identifiability of the underlying law (Moradi et al., 2024). Inference about governing equations therefore requires diagnostics beyond rollout loss, including attractor geometry, invariant measures, and problem-specific structural constraints.

Open questions remain broad. They include formal characterizations of when deeper KANs induce smoother inner representations; scalable implementations for very high-dimensional domains such as language modeling; principled uncertainty quantification beyond mean-field approximations; adaptive basis and knot selection with controllable computational overhead; and structural theory for hybrid additive–multiplicative, geometry-aware, or kernelized KAAM variants (Liu et al., 2024, Liu et al., 24 Sep 2025, Ju, 4 Dec 2025). The current literature indicates that KAAM is not a single settled model but a theorem-backed design space whose most useful forms depend on whether the task prioritizes expressivity, interpretability, statistical efficiency, computational tractability, or explicit domain structure.