Autoregressive-Weight-Enhanced AR-KAN

• The paper introduces a hybrid AR-KAN framework that leverages autoregressive weighting for memory extraction and KAN for adaptive nonlinear modeling, resulting in improved prediction accuracy.
• It employs a two-stage architecture where the AR module filters and condenses temporal data, while the KAN flexibly models high-frequency nonlinear interactions.
• Empirical evaluations show AR-KAN outperforms conventional approaches on 72% of real-world datasets, proving its effectiveness in forecasting almost periodic signal regimes.

Autoregressive-Weight-Enhanced AR-KAN refers to a hybrid modeling framework for time series forecasting that integrates a pre-trained autoregressive (AR) memory module with a Kolmogorov-Arnold Network (KAN), justified by the Universal Myopic Mapping Theorem. This approach is designed to explicitly address limitations of both classical statistical and neural architectures—specifically in scenarios involving almost periodic functions with incommensurate frequency components—by combining the spectral properties of AR models and the nonlinear expressiveness of KANs. The weighting enhancement stems from the use of autoregressive coefficients for memory extraction, retaining only the most informative inputs and eliminating redundancy in dynamic forecasting.

1. Theoretical Foundations and Definitions

Autoregressive-Weight-Enhanced AR-KAN is motivated by the Universal Myopic Mapping Theorem, which posits that any shift-invariant, myopic dynamical system can be approximated by an architecture consisting of a bank of linear filters followed by a static nonlinearity. In this context:

• The AR component acts as a memory module, extracting relevant features from recent time series observations through a set of learned linear weights:

$\hat{x}(n+1) = \sum_{i=0}^{p-1} a_i x(n-i)$

where $\{a_i\}$ are AR coefficients fitted (e.g., via Yule–Walker equations), $p$ is the order, and $x$ is the input sequence.

• The static nonlinear mapping is realized by a Kolmogorov-Arnold Network (KAN), in which classical fixed-activation neurons are replaced by learnable univariate functions (splines), enabling highly flexible nonlinear modeling while retaining the relative simplicity of shallow architectures.

This structure yields a two-stage predictor:

$$\hat{x}_{n+1} = \mathrm{KAN}(a_0 x_n, a_1 x_{n-1}, \ldots, a_{p-1} x_{n-p+1})$$

The AR module's optimal weights result from minimizing the forecast error under the autocorrelation structure $R$ of the data:

$$a = R^{-1} r \quad \text{where} \quad R = [r(|i-j|)], \quad r(i) = \mathbb{E}[x(n) x(n-i)]$$

2. Model Architecture and Weight Enhancement

The AR-KAN framework consists of two serial components:

1. AR Memory Module: The input sequence is filtered by AR weights, which compress temporal information and emphasize components most useful for immediate prediction. The role of weighting is to maximize the retention of predictive information while minimizing redundancy. The AR memory bank acts as a set of linear predictors, each corresponding to specific lags. The coefficients are data-driven and can be interpreted as impulse responses $h_i(n) = a_i \delta(n-i)$.
2. KAN Static Nonlinear Module: The filtered outputs are passed through a KAN, which replaces fixed activations with learnable splines. This allows the network to approximate highly nonlinear, high-frequency interactions that arise in almost periodic signals with incommensurate frequencies—a regime where conventional neural networks with low-frequency bias or even Fourier Neural Operators may fail.

Compared to single-model approaches (MLP, LSTM, ARIMA, FNO), AR-KAN’s autoregressive enhancement ensures that only the most relevant past information is retained and processed by the nonlinear mapping, reducing overfitting and mitigating the curse of dimensionality.

3. Mathematical Formulation and Optimization Principles

The general forecasting problem is formalized as:

$$\hat{x}_{n+1} = \mathcal{A}(x_n, x_{n-1}, \ldots, x_{n-p+1})$$

where in AR-KAN:

• $\mathcal{A}$ combines the AR memory output and the nonlinear KAN mapping.
• AR weights are computed by solving the covariance-based normal equations:

$a = R^{-1} r$

• The memory stage optimizes a quadratic objective over the past $p$ lags:

$$L = \sum_{i=0}^{p-1} \mathbb{E}[y_i(n) x(n+1)] - \frac{1}{2} \mathbb{E}\left[ \left( \sum_{i=0}^{p-1} y_i(n) \right)^2 \right],\quad y_i(n) = w_i x(n-i)$$

The solution $w^* = R^{-1} r$ guarantees information-theoretic efficiency.

The KAN, operating after memory extraction, is not limited to polynomial or trigonometric activations, hence allowing flexible fitting of functions generated by almost periodic signals.

4. Comparison to Conventional Time Series Models

Autoregressive-Weight-Enhanced AR-KAN aims to overcome fundamental limitations in both statistical and neural time series methods:

• ARIMA: Well-suited for periodic or almost periodic signals due to its spectral bias. However, ARIMA lacks the capacity to model nonlinear interactions, leading to suboptimal performance where nonlinearities are significant.
• Standard Neural Networks: MLPs, LSTMs, and Transformer variants often impose a low-frequency bias, causing loss of high-frequency (fine-structure) information, particularly problematic for signals with complex periodicities.
• Fourier Neural Networks (e.g., FAN, FNO): Effective for strictly periodic phenomena; nevertheless, the superposition of incommensurate frequencies yields an almost periodic signal, not strictly periodic, which complicates the efficacy of fixed-frequency basis approaches.

Autoregressive weighting (the “enhancement”) provides information-optimal short-term memory, while the KAN delivers high-fidelity nonlinear approximation. Empirical evaluations show AR-KAN delivers superior results on 72% of tested real-world datasets, demonstrating performance parity with ARIMA on noisy, almost periodic functions and outperforming state-of-the-art neural architectures.

5. Practical Scenarios and Demonstrated Performance

Empirical studies have evaluated AR-KAN on tasks ranging from synthetic, noisy almost periodic functions to 18 real-world datasets varying in origin (e.g., economics, meteorology, healthcare). In these settings, AR-KAN’s architecture consistently surpasses alternatives:

• In almost periodic regimes (superpositions of sines with incommensurate frequencies), AR-KAN matches ARIMA in accuracy, outperforming both classical and neural baselines.
• Across the real-world dataset suite, AR-KAN achieves best-in-class results on approximately 72% of tasks. The method reliably balances spectral and nonlinear generalization—this is particularly notable where signals exhibit both linear and complex interaction components.
• The ability of the KAN to model high-frequency residuals and nonlinearities assigns AR-KAN a unique advantage over strictly linear or Fourier-based systems.

6. Extensions, Related Techniques, and Potential Integrations

The separation of memory extraction and nonlinear processing in AR-KAN aligns with recent advances in hybrid inference systems and filter design:

• Methods for stochastic unknown input realization via AR modeling and least squares (as in (Yu et al., 2014)) similarly use output autocorrelations and input modeling to augment estimation methodology (e.g., Kalman filters with AR-based unknown input models).
• The AR-KAN approach could be extended to high-dimensional states using model order reduction (ROM-based filtering (Yu et al., 2014)) or graph-valued data using GNN-based AR models for dynamic topologies (Zambon et al., 2019).
• Integration with state estimation for systems with colored disturbances is a plausible application area, leveraging the AR memory component as an explicit realization of temporal correlation (disturbance modeling in augmented Kalman filtering).

A plausible implication is that the autoregressive weighting enhancement in AR-KAN could be further adapted to scenarios where disturbance statistics are unknown or evolving, employing data-driven recovery of input statistics and corresponding AR weights.

7. Significance and Current Research Directions

Autoregressive-Weight-Enhanced AR-KAN represents an overview of classical linear prediction and modern nonlinear approximation, underpinned by rigorous theorems and validated on complex time series forecasting problems. Its structured approach—first isolating short-term spectral memory through optimal AR weighting, then enacting adaptive nonlinear transformation via KAN—addresses key deficits in both neural and statistical networks.

Ongoing research is likely to refine KAN modules for richer nonlinear transformations, explore integration with graph-structured data, and develop adaptive weighting schemes for non-stationary settings, as well as investigate connections to innovations-based filtering and ROMs in high-dimensional dynamical systems. The method’s theoretical generality and demonstrated performance across diverse datasets suggest its broad applicability in advanced forecasting and system identification.