Temporal Kolmogorov-Arnold Networks

Updated 12 January 2026

Temporal Kolmogorov-Arnold Networks are neural architectures that apply learnable spline functions to decompose and model sequential, time-dependent data.
They replace fixed linear weights with flexible spline mappings in recurrent models, achieving state-of-the-art performance in forecasting, classification, and anomaly detection.
T-KAN enables interpretability through symbolic regression of spline coefficients and supports hardware optimizations for rapid, efficient temporal modeling.

Temporal Kolmogorov-Arnold Networks (T-KAN) are neural architectures that extend the Kolmogorov–Arnold Network paradigm to sequential, time-dependent data. Rooted in the Kolmogorov–Arnold representation theorem, T-KAN architectures replace fixed linear weights and activation functions found in classical models—such as LSTMs and MLPs—with learnable, spline-parameterized or basis-expansion edge functions. This enables interpretable, parameter-efficient modeling of complex temporal and dynamical systems, achieving state-of-the-art results in time series forecasting, classification, anomaly detection, and hidden-physics discovery.

1. Mathematical Foundations and Theoretical Rationale

The Kolmogorov–Arnold representation theorem guarantees that any continuous multivariate function $f: [0,1]^n \to \mathbb{R}$ can be exactly expressed as a finite sum of outer functions applied to sums of univariate inner functions: $f(x_1, \dots, x_n) = \sum_{q=1}^{2n+1} \Phi_q\Bigl(\sum_{p=1}^n \varphi_{q,p}(x_p)\Bigr),$ where $\varphi_{q,p}$ and $\Phi_q$ are continuous univariate maps (Xu et al., 2024). T-KAN leverages this construction in neural architectures by parameterizing each edge function between network nodes as a learnable spline—commonly cubic B-splines, radial basis functions, or even truncated Fourier components for certain applications (Zhou et al., 2024).

For temporal modeling, the theorem guides the decomposition of a time series window $x_{0:i} = (x_{i-T+1}, ..., x_{i}) \in \mathbb{R}^T$ such that the network applies distinct 1D splines or basis functions on each lag, followed by summations and outer nonlinearities. In continuous-time cases (e.g., neural ODEs), T-KAN incorporates time explicitly as an additional input, yielding functions of both state and time, $f(x, t)$ (Koenig et al., 2024).

2. Core Architecture: Spline-parametric Recurrence and Gating

T-KAN architectures generalize classical recurrent cells (e.g., LSTM, GRU) by replacing affine weight-matrix transforms with sums of univariate spline mappings. In high-frequency limit-order-book forecasting, each gate in an LSTM cell—input, forget, output, candidate—is computed as: $g_t = \varphi^g([h_{t-1}, x_t]),$ where $\varphi^g(\cdot)$ is a sum of B-spline basis expansions learned per coordinate. The full cell update equations become (Makinde, 5 Jan 2026): $\begin{aligned} i_t &= \sigma(\mathrm{KAN}_i([h_{t-1}, x_t])), \ f_t &= \sigma(\mathrm{KAN}_f([h_{t-1}, x_t])), \ \tilde{g}_t &= \tanh(\mathrm{KAN}_g([h_{t-1}, x_t])), \ o_t &= \sigma(\mathrm{KAN}_o([h_{t-1}, x_t])), \ c_t &= f_t \odot c_{t-1} + i_t \odot \tilde{g}_t, \ h_t &= o_t \odot \tanh(c_t). \end{aligned}$ Learnable spline coefficients are regularized for smoothness and trained by backpropagation (Genet et al., 2024). In multi-layer KANs, this edge-wise spline structure is composed recursively and admits direct visualization, symbolic regression, and pruning.

RKAN (Recurring KAN) layers further augment the basic edge-wise function with per-node memory vectors, updating them in a linear fashion and folding them into the KAN composition at each time step (Genet et al., 2024, Genet et al., 2024, Xu et al., 2024).

3. Temporal Extensions and Model Integration

T-KAN has been embedded in a variety of temporal modeling frameworks:

Sequence Forecasting: Standalone T-KAN or T-KAN heads on convolutional/recurrent/tranformer models achieve notable gains on time series (e.g., FI-2010 LOB forecasting, univariate/multivariate financial series) (Makinde, 5 Jan 2026, Xu et al., 2024).
Neural ODEs: KAN-ODE frameworks treat time as an explicit input coordinate and leverage radial basis expansions for enhanced accuracy and grid independence (Koenig et al., 2024).
Mixer/Transformer Hybrids: TSKANMixer integrates depth-2 KAN layers into MLP-mixer blocks, achieving 19–31% MSE reduction on key multivariate datasets (Hong et al., 25 Feb 2025). TKAT swaps every LSTM cell for T-KAN in an encoder–decoder transformer, aligning Kolmogorov–Arnold memory cells with self-attention mechanisms (Genet et al., 2024).
Physics-informed Models: In quantum dynamical tasks, chain-of-KAN architectures enforce strict causality and are trained under additional physics-informed loss terms (Ehrenfest constraints), reducing data requirements by nearly 20× over TCNs (Sen et al., 23 Sep 2025).

4. Empirical Results and Benchmark Comparisons

Comprehensive benchmark studies demonstrate T-KAN's efficacy:

On high-frequency LOB data, T-KAN achieves a 19.1% improvement in F1-score at $k=100$ step horizon over DeepLOB baselines, and generates 132.48% return versus -82.76% drawdown under transaction costs (Makinde, 5 Jan 2026).
In KAN-ODEs, T-KAN converges to Lotka–Volterra solutions with MSE $<8.3\times10^{-7}$ , outperforming tanh-MLP ODEs at similar parameter counts (Koenig et al., 2024).
For time-series classification on 128 UCR benchmarks, T-KAN matches or exceeds classical MLP in accuracy and F1 (mean $\sim$ 82.3%), with improved adversarial robustness owing to lower Lipschitz constants (Dong et al., 2024).
TSKANMixer improves TSMixer’s MSE by up to 31% on benchmark datasets but incurs considerable computational overhead due to spline evaluation (Hong et al., 25 Feb 2025).
Physics-informed T-KANs achieve $R^2 > 0.95$ with only $5.4\%$ as many samples as comparable temporal convolutional networks (Sen et al., 23 Sep 2025).

5. Interpretability, Concept Drift, and Symbolic Extraction

A central feature of T-KAN is the interpretability of edge-wise splines:

Learned B-spline gate activations visibly display “dead-zones” for filtering microstructure noise, with steep flanks amplifying genuine temporal shifts. Asymmetric gates can reflect nuanced sensitivities, such as bid/ask imbalances in trading (Makinde, 5 Jan 2026).
Time-domain concept drift detection is realized by monitoring changes in spline coefficients over sliding windows; significant norm changes or functional clustering are flagged as drift events (Xu et al., 2024).
Symbolic regression tools (e.g., Eureqa, PySR) fit closed-form algebraic expressions to each learned spline, yielding transparent formulas for forecasts and basis contributions (Xu et al., 2024, Koenig et al., 2024).
In KAN-ODE applications, post-training symbolic regression can recover underlying physical source terms with high fidelity (e.g., Fisher–KPP reaction term) (Koenig et al., 2024).

6. Hardware Optimization and Computational Efficiency

Due to their construction as per-coordinate univariate spline evaluations, T-KAN cells are suited for hardware optimization:

FPGA synthesis via HLS allows high-throughput ( $<$ 1 μs per cell update), with resource usage dominated by BRAM for spline knots and DSP slices for basic arithmetic (Makinde, 5 Jan 2026).
Inference speed and model compactness are highlighted in anomaly detection (KAN-AD), where truncated Fourier basis replaces splines, yielding models with order-of-magnitude fewer parameters and up to $55\times$ faster inference (Zhou et al., 2024).
The parameter efficiency and grid flexibility of T-KAN have been demonstrated in scientific ML contexts—where rapid scaling ( $\mathcal{O}(N^{-4})$ error reduction) facilitates sharp feature learning with fewer parameters (Koenig et al., 2024).

7. Limitations, Extensions, and Research Directions

T-KAN presents several computational and modeling challenges:

Spline parameterization increases computational load (training time up to $50\times$ that of MLPs for comparable width) and induces challenging hyperparameter tuning (grid size, spline order, hidden dimension) (Hong et al., 25 Feb 2025).
Overly flexible spline grids can lead to degenerate fits or optimization challenges; ablation studies confirm that simple base activations (e.g., Silu+linear components) often dominate discriminative power (Dong et al., 2024).
In high-dimensional or noisy environments, optimization strategies and hybrid models are required to maintain performance (Somvanshi et al., 2024).

Active research is focused on:

Efficient spline implementations and lookup-based approximations.
Convolutional and attention-based hybrid architectures (e.g., convolutional-KANs, attention-KANs).
Extension to multivariate and multi-output time-series.
Unsupervised pretraining and contrastive learning on temporal libraries.
Symbolic regression and physical interpretation in dynamical systems.

In summary, Temporal Kolmogorov-Arnold Networks constitute a technically rigorous, interpretable, and empirically validated framework for sequential modeling. Grounded in mathematical superposition theory, they demonstrate notable advances in predictive accuracy, explainability, robustness, and hardware deployment across finance, scientific ML, anomaly detection, and dynamical systems modeling (Makinde, 5 Jan 2026, Xu et al., 2024, Koenig et al., 2024, Hong et al., 25 Feb 2025, Dong et al., 2024, Sen et al., 23 Sep 2025).