---
title: 'KAN-IDIR: Multi-Disciplinary Insights'
url: https://www.emergentmind.com/topics/kan-idir
type: topic
---

# KAN-IDIR: Multi-Disciplinary Insights

KAN-IDIR is a domain-dependent research label rather than a single standardized concept. One explicit formal usage names KAN-IDIR and RandKAN-IDIR as KAN-based implicit neural representations for deformable image registration [2509.22874]. The same string also appears as a shorthand for “KAN-based Intrusion Detection with Resource constraints via DKD” in industrial control system security [2603.03486], as an abbreviation for Kan injectivity and induced KZ-pseudomonads in 2-category theory [2211.00380], and as a mnemonic organizing several KAN-centered summaries in equation discovery, phase-transition identification, quantum-inspired KAN design, and kaon identification [2411.11801] [2503.17996] [2510.08650] [1611.07394]. The term therefore designates a family of unrelated constructions unified only by local disciplinary usage.

## 1. Terminological scope

The label is used across several technically distinct literatures. In medical image analysis, it denotes an instance-specific DIR method built from KAN-based INRs. In industrial cybersecurity, it denotes a teacher–student IDS pipeline in which a high-capacity KAN transfers knowledge to a tiny MLP via DKD. In higher category theory, it denotes a theory of Kan injectivity and the KZ-pseudomonads induced by a set of 1-cells. Other appearances are mnemonic rather than canonical, but they preserve the same pattern of attaching “KAN” to an application-specific expansion.

| Usage | Core meaning | Representative paper |
|---|---|---|
| Medical image registration | KAN-IDIR and RandKAN-IDIR for deformable registration | [2509.22874] |
| ICS intrusion detection | “KAN-based Intrusion Detection with Resource constraints via DKD” | [2603.03486] |
| 2-category theory | Kan injectivity and induced KZ-pseudomonads | [2211.00380] |
| Other mnemonic uses | KAN summaries in equation discovery, phase transitions, QuIRK, kaon ID | [2411.11801], [2503.17996], [2510.08650], [1611.07394] |

This multiplicity matters because the same string spans two different meanings of “KAN”: Kolmogorov–Arnold networks in machine learning and Kan extensions or Kan injectivity in category theory. A plausible implication is that any technical reading of KAN-IDIR must be anchored to its source paper rather than inferred from the acronym alone.

## 2. KAN-IDIR in deformable image registration

In deformable image registration, KAN-IDIR and RandKAN-IDIR are introduced as the first integration of Kolmogorov-Arnold Networks into DIR with implicit neural representations [2509.22874]. The problem is posed by seeking a continuous displacement field $u(x)\in\mathbb{R}^d$ on a domain $\Omega\subset\mathbb{R}^d$, with transformation
$$
\phi(x)=x+u(x).
$$
The instance-specific objective combines image similarity, regularization, and a Jacobian-based anti-folding penalty:
$$
L(\phi)=L_{\mathrm{sim}}(I_f,I_m\circ\phi)+\lambda R(u)+\beta P(J_\phi),
$$
with $J_\phi(x)=\nabla\phi(x)=I+\nabla u(x)$.

The default similarity term is the negative normalized cross-correlation computed over minibatches of coordinates. Regularization is total variation on the displacement gradient,
$$
R_{\mathrm{TV}}(u)=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\|\nabla u(p)\|_1,
$$
and folding reduction is handled by
$$
P_{J\det}=\frac{1}{|\Omega_p|}\sum_{p\in\Omega_p}\max(0,-|J_\phi(p)|+\epsilon),
$$
with $\epsilon=0.1$. The reported loss is
$$
L=L_{\mathrm{sim}}+\lambda L_{\mathrm{smooth}}+\gamma L_{J\det},
$$
with $\lambda=0.4$ and $\gamma=15$.

The network is a Chebyshev-KAN INR with two hidden layers of width $70$. A single layer computes
$$
[y(x)]_{b,o}=\sum_{i=1}^{N_{\mathrm{in}}}\sum_{d\in\mathcal{D}} T_d(\tilde{x}_{b,i})\cdot C_{i,o,d},
$$
where $T_n(z)=\cos(n\arccos z)$ are Chebyshev polynomials and inputs are normalized to $[-1,1]$ via $\tanh$. Each layer also includes a learnable linear skip connection,
$$
L_m(x)=W_m\cdot b(x)+y_m(\tilde{x}),
$$
with $b(\cdot)$ given as SiLU. The full INR is a composition of these layers, and its output head maps coordinates to $u(x)$.

KAN-IDIR uses a sequential basis set $\mathcal{D}=\{0,1,\dots,D\}$ shared across layers. RandKAN-IDIR instead samples
$$
\mathcal{D}_m=\{0\}\cup\mathbb{S}_m,\qquad \mathbb{S}_m\sim \text{Uniform $k$-sample without replacement from }\{1,2,\dots,K\},
$$
which reduces the number of evaluated basis functions while preserving diversity across layers. The main experiments use $k=12$ and $K=84$.

The optimization is instance-specific rather than dataset-trained. The model is initialized randomly and trained with Adam at learning rate $10^{-4}$ for $1500$ iterations, keeping the learning rate constant for the first $50\%$ of iterations and then using cosine annealing. Each iteration samples $10{,}000$ points from an organ mask or application-specific spatial subset. This places KAN-IDIR between classical iterative registration and dataset-trained CNN or transformer pipelines: it avoids training on large cohorts, but still requires per-pair optimization.

## 3. Empirical behavior of KAN-IDIR and RandKAN-IDIR

The reported evaluation spans lung CT, brain MRI, and cardiac MRI, with comparisons to other INR methods, dataset-trained deep learning models, and classical registration [2509.22874]. On DIR-Lab, performance is measured by TRE; on OASIS-1 and ACDC, by DSC, HD95, and NJD.

| Dataset | KAN-IDIR | RandKAN-IDIR |
|---|---|---|
| DIR-Lab | TRE $0.98\pm1.10$ mm; $63.3$ s; $2.2$ GB | TRE $0.99\pm1.10$ mm; $43.1$ s; $1.4$ GB |
| OASIS-1 | DSC $0.793\pm0.16$; HD95 $1.96\pm0.32$ mm; NJD $0.048\%\pm0.03\%$ | DSC $0.792\pm0.17$; HD95 $1.97\pm0.32$ mm; NJD $0.049\%\pm0.04\%$ |
| ACDC | DSC $0.814\pm0.09$; HD95 $7.39\pm3.17$ mm; NJD $2e^{-5}\%\pm2e^{-4}\%$ | DSC $0.810\pm0.09$; HD95 $7.88\pm3.28$ mm; NJD $4e^{-5}\%\pm2e^{-4}\%$ |

On DIR-Lab, the outlier distribution for KAN-IDIR is reported as $60.13\%$, $81.69\%$, $98.14\%$, and $1.86\%$ for thresholds $\leq1$ mm, $\leq2$ mm, $\leq3$ mm, and $>3$ mm; RandKAN-IDIR reports $60.28\%$, $81.43\%$, $98.09\%$, and $1.91\%$. Classical pTV attains $0.95\pm1.15$ mm TRE but requires approximately $442$ s, while IDIR requires $260.8$ s and ccIDIR $92.6$ s. This makes the KAN variants competitive among INR methods while remaining substantially faster than pTV.

The paper emphasizes seed stability. Across $10$ random seeds on DIR-Lab, the difference in $>3$ mm outlier counts between best and worst seed is $4$ for KAN-IDIR versus $56$ for the SIREN-based baseline. The reported interpretation is that KAN-based coordinate-to-displacement maps are less seed-sensitive than MLP or SIREN INRs.

RandKAN-IDIR is also compared with an adaptive index-learning variant, A-KAN-IDIR. RandKAN-IDIR is reported to slightly outperform A-KAN-IDIR while eliminating its additional training-time complexity, with approximately $20\%$ runtime reduction and approximately $20\%$ less VRAM in the reported tests. This makes randomized basis sampling the preferred sparse-basis variant in the study.

The accuracy–efficiency trade-off is described as nearly linear in basis count. Increasing $D$ in KAN-IDIR or $k$ in RandKAN-IDIR improves accuracy but scales runtime and memory with $|\mathcal{D}|$. On DIR-Lab, TRE improves up to about $D=28$ and then plateaus. For RandKAN-IDIR, randomized sampling with $k=12$, $K=84$ is reported to exceed the sequential choice $k=12$, $K=12$ by $+1.5\%$ DSC and to match $k=24$, $K=24$ while running faster and using less memory.

## 4. KAN-IDIR as intrusion detection under resource constraints

In industrial control system security, KAN-IDIR denotes a lightweight intrusion-detection framework that uses a high-capacity KAN teacher and a tiny MLP student trained by decoupled knowledge distillation [2603.03486]. The target setting is binary cyber-attack detection for SWaT and WADI, both operating at about $1$ Hz sampling and involving multivariate sensor and actuator streams. The method is explicitly designed for edge and real-time CPS environments in which model size, inference latency, and memory use are critical.

The teacher is a KAN modeled through the Kolmogorov–Arnold representation
$$
f(X)=KAN(X)=\sum^{2T+1}_{q=1}\Phi_q\Bigl(\sum^T_{p=1}\phi_{q,p}(X_p)\Bigr),
$$
with univariate edge activations defined as
$$
\phi(X)=\omega_b \mathrm{SiLU}(X)+\omega_s \mathrm{Spline}(X),\qquad
\mathrm{Spline}(X)=\sum^{G+K}_{i=1} c_i B_i(X).
$$
The reported KAN parameter count is
$$
\mathrm{Parameters}=(d_{\mathrm{in}}\times d_{\mathrm{out}})(G+K+3)+d_{\mathrm{out}}.
$$
Teacher settings are $K=1$, $G=50$ for WADI and $K=3$, $G=50$ for SWaT.

The student is a compact MLP trained from hard labels and a DKD objective. The decomposition
$$
\mathrm{KD}=\mathrm{TCKD}+(1-p_t^T)\cdot \mathrm{NCKD},\qquad
\mathcal{L}_{\mathrm{DKD}}=\alpha\cdot\mathrm{TCKD}+\beta\cdot\mathrm{NCKD}
$$
separates target-class and non-target-class transfer. The total loss uses a warm-up schedule blending hard-label loss with DKD. Reported hyperparameters are $w_{\mathrm{warmup}}=80$, $\lambda=0.1$, $\alpha=5$, $\beta=1$ for SWaT, and $w_{\mathrm{warmup}}=5$, $\lambda=0.2$, $\alpha=5$, $\beta=1$ for WADI. Inputs are standardized by
$$
X_{\mathrm{scaled}}=\frac{X-\mu}{\sigma}.
$$

The resulting student models are extremely small: $1{,}622$ parameters for SWaT and $2{,}522$ for WADI. On SWaT, the bare student reports Acc $99.05$, Prec $98.59$, Rec $84.96$, F1 $91.27$, whereas DKD-MLP reports Acc $99.49$, Prec $98.90$, Rec $92.24$, F1 $95.45$, a gain of $+4.18$ F1 points. On WADI, the bare student reports Acc $99.48$, Prec $97.21$, Rec $93.62$, F1 $95.38$, whereas DKD-MLP reports Acc $99.82$, Prec $98.88$, Rec $98.02$, F1 $98.45$, a gain of $+3.07$ F1 points. The WADI KAN teacher has $198{,}750$ parameters and reports Acc $99.88$, Prec $98.84$, Rec $99.09$, F1 $98.96$.

The framework uses per-instance inference with no sliding windows and only standardization preprocessing. The reported memory footprint is approximately $6.3$ KB for the SWaT student and approximately $9.9$ KB for the WADI student, assuming float32 weights. Estimated compute per instance is approximately $1.59$k MACs for SWaT and approximately $2.50$k MACs for WADI. Throughput, latency, and energy are not measured in the paper, but the student is explicitly positioned for PLC or edge-gateway deployment. The paper also notes a reporting inconsistency for SWaT: a results table labels the teacher as an “MLP teacher,” while the narrative consistently describes a KAN teacher.

## 5. KAN-IDIR in 2-category theory

In category theory, KAN-IDIR denotes Kan injectivity and induced KZ-pseudomonads [2211.00380]. The setting is an adequate 2-category $\mathcal{K}$ that is locally small, has the required small weighted bicolimits, and satisfies a smallness hypothesis ensuring convergence of transfinite pseudochains. The central object is left Kan injectivity with respect to a set $\mathcal{H}$ of 1-cells.

For $h:A\to A'$ and $f:A\to X$, a left Kan extension of $f$ along $h$ is a pair $(f/h,\xi_f)$ with $f/h:A'\to X$ and $\xi_f:f\Rightarrow (f/h)\circ h$ satisfying the usual universal property. Equivalently, precomposition with $h$ induces an adjunction on hom-categories, with left adjoint $(-)/h$. The universal property is expressed by the natural isomorphism
$$
\mathcal{K}(B,X)(\mathrm{Lan}_h f,g)\cong \mathcal{K}(A,X)(f,g\circ h).
$$

Two forms of injectivity are distinguished. Weak left Kan injectivity of $X$ with respect to $\mathcal{H}$ requires existence of the relevant left adjoints $\mathcal{K}(A,X)\to\mathcal{K}(A',X)$, or equivalently that $\mathcal{K}(h,X)$ be a right adjoint. Strong left Kan injectivity further requires the canonical Kan 2-cell $\xi_f$ to be invertible. In the presence of bicocomma objects, weak and strong notions coincide after saturation, written $\mathrm{WLInj}(\mathcal{H})=\mathrm{LInj}(\bar{\mathcal{H}})$ for a canonical closure $\bar{\mathcal{H}}$.

The induced KZ-pseudomonad arises from a transfinite pseudochain. For each object $X$, one builds $(X_i)_{i\in \mathrm{Ord}}$ by inserting 1-dimensional Kan-extension data via wide bipushouts and 2-dimensional universality constraints via bicoequinserters or bicoequifiers, taking bicolimits at limit ordinals. Choosing a sufficiently large regular cardinal $\kappa$, one sets
$$
T_{\mathcal{H}}X:=X_\kappa,\qquad \eta_X:=x_{0,\kappa}:X\to X_\kappa.
$$
By Marmolejo–Wood’s left Kan pseudomonad construction, this yields a KZ-pseudomonad $T_{\mathcal{H}}=(T_{\mathcal{H}},\eta,\mu)$.

The main theorem identifies the pseudoalgebras of $T_{\mathcal{H}}$ with the objects left Kan injective with respect to $\mathcal{H}$, and the pseudoalgebra morphisms with the morphisms preserving Kan extensions along $\mathcal{H}$. A pseudoalgebra structure map $a_X:T_{\mathcal{H}}X\to X$ satisfies
$$
a_X\circ \eta_X\cong 1_X,\qquad a_X\circ T_{\mathcal{H}}(a_X)\cong a_X\circ \mu_X.
$$
Examples in $\mathbf{Cat}$ recover familiar cocompletion doctrines. With appropriate choices of $\mathcal{H}$, one obtains finite-colimit completion and links to Garner–Lack’s $\Phi$-lex-colimit KZ-pseudomonads. In this literature, KAN-IDIR has no connection to Kolmogorov–Arnold networks; it is entirely about Kan extensions, injectivity, and lax-idempotent 2-monadic structure.

## 6. Related KAN-IDIR-style usages in adjacent literatures

Several other papers use the string as a mnemonic rather than as the official title of a method. These uses are technically separate but are informative because they emphasize recurring themes of interpretability, sparsity, and structure discovery.

In nonlinear equation discovery, KAN/MultKAN combined with Sequentially Regularized Derivatives for Denoising and Physics-Informed Spline Fitting forms a pipeline for identifying governing ODEs and PDEs from noisy data [2411.11801]. The framework proceeds through SRDD denoising, KAN-guided library construction, and PISF-based term elimination. It was tested on the forced Duffing oscillator, Van der Pol oscillator, Burgers’ equation, and the Bouc–Wen model; it converged to the true equation for the first three and produced an approximate hysteresis-capable model for Bouc–Wen. The summary explicitly attributes the framework to Pal et al. (2024). A plausible implication is that the mnemonic use of KAN-IDIR here foregrounds interpretable discovery rather than a fixed acronym expansion.

In statistical physics, a KAN-based classifier is trained directly on raw site-percolation and Ising configurations [2503.17996]. The reported architecture uses input size $L\times L$, a hidden layer of $5$ nodes, and an output layer of $2$ nodes, with edge activations $\psi(x)=\mathrm{SiLU}(x)+\mathrm{B\text{-}spline}(x)$ and regularization based on activation $L_1$ norm and entropy. At $200$ training steps, reported accuracies reach $0.995$, $0.994$, $0.998$, and $0.997$ for Ising at $L=10,20,30,40$, and $0.992$, $0.991$, $0.998$, and $0.999$ for percolation. Finite-size scaling of the learned output yields $\nu\approx1.01$ for the Ising model and $\nu\approx1.33$ for percolation. Here the string is used to organize a KAN-based identification program rather than to name a separate architecture family.

In "QuIRK: Quantum-Inspired Re-uploading KAN," the explanatory label “KAN-IDIR” is expanded as Kolmogorov–Arnold Networks, Data Re-uploading, Interpretability, and closed-form Derivations [2510.08650]. QuIRK replaces B-spline edge functions with single-qubit data re-uploading models, so each edge function becomes a trigonometric polynomial obtained from a single-qubit expectation. A representative expression is
$$
g(x;\Theta)=\langle 0|U(x;\Theta)^\dagger ZU(x;\Theta)|0\rangle,
$$
and the paper states that QuIRK retains interpretability and closed-form solutions while often using fewer parameters than classical KANs, especially for periodic targets.

A further mnemonic usage appears in the NA62 kaon-identification literature, where the supplied summary maps KAN-IDIR to “kaon identification in-rate differential Cherenkov ring tagging” [1611.07394]. The underlying system is the CEDAR counter upgraded to KTAG, operating in a $750$ MHz unseparated hadron beam with a kaon component of about $6\%$. The reported performance includes kaon-tagging efficiency exceeding $98\%$ with at least $5$ sectors in coincidence, pion misidentification of $O(10^{-4})$, and overall event-time resolution of about $70$ ps. This usage is terminologically incidental, but it shows that KAN-IDIR can also be repurposed outside the Kolmogorov–Arnold and categorical contexts.

Across these variants, the string functions less as a stable acronym than as a movable label attached to structure-aware inference or identification. This suggests that “KAN-IDIR” is best treated bibliographically as a context-sensitive term whose meaning is determined entirely by the paper in which it appears.

Source: https://www.emergentmind.com/topics/kan-idir