🤖 machine learning

Temporal Functional Circuits: From Spline Plots to Faithful Explanations in KAN Forecasting

This contribution introduces Temporal Functional Circuits, a framework based on gated residual Kolmogorov-Arnold networks (KANs) that transforms explicit edge functions into trustworthy, time-grounded explanations for time series forecasting, thereby achieving competitive performance and demonstrating the predictive value of learned spline shapes.

Original authors: Naveen Mysore

Published 2026-05-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Naveen Mysore

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for tomorrow. You have a super-smart computer model that analyzes past data to make its prediction. Usually, these models are like "black boxes": they provide an answer, but if you ask why they chose that answer, they merely say: "I processed a lot of numbers."

This article introduces a new method for building such prediction models using something called KANs (Kolmogorov-Arnold Networks). Instead of being a black box, a KAN is like a transparent machine where every single wire carries a label.

Here is the breakdown of what the article accomplishes, using simple analogies:

1. The Problem: "Pretty Pictures" Are Not Enough

In the past, researchers tried to explain KANs by drawing images of the mathematical functions inside the wires (so-called "splines"). It is like looking at a blueprint of a car engine and saying: "See these curves? That's how it works!"

But the article argues that merely seeing the curve does not reveal whether the engine actually runs or whether it is even necessary. Sometimes a model draws a fancy curve on a wire but completely ignores it. The article calls this "visual plausibility" versus "reliability." Just because a graphic looks like it explains the prediction does not mean it actually caused the prediction.

2. The Solution: The "Smart Switch" (Gated Residual KAN)

To fix this, the authors built a new model with a Smart Switch (a "Gate").

The Linear Part (The Base): Imagine a simple, reliable calculator that handles the straightforward parts of the prediction (like "it is usually warmer in summer").
The KAN Part (The Correction): Imagine a fancy artist who adds the complex, wavy details (like "but today there is a strange storm front").
The Gate: This is a switch that decides for each individual data point whether the artist is allowed to work or if we should just stick with the calculator.

If the data is simple, the switch stays closed, and the model uses the simple calculator. If the data is chaotic and complex, the switch opens, and the artist (the KAN) steps in to correct the prediction.

3. The "Temporal Functional Circuit" (The Detective Work)

The article presents a framework called Temporal Functional Circuits. Think of this as a detective kit for the model. Instead of just looking at the wires, the detective asks three specific questions for every single wire in the network:

Who is this wire talking to? (Lag Assignment): Does this wire care about what happened 3 hours ago or 3 days ago? The article assigns the wire directly to specific time points.
How important is this wire? (Importance): The article ranks the wires. Some wires are "superstars" doing heavy lifting; others are just decoration. They found that wires with the largest "movement radius" (the most complex curves) are usually the most important.
What happens if we cut the wire? (Intervention): This is the most important part. The researchers literally "cut" (set to zero) the top wires and observed the prediction.
- The Result: When they cut the top wires, the prediction got worse. This proves that these wires were actually doing the work and not just looking good.
- The "Spline" Test: They also tried removing only the "curve-rich" part of the wire while keeping the straight part. The prediction still got worse. This proves that the specific shape of the curve matters, not just the fact that the wire exists.

4. What They Found

The researchers tested this on eight different real-world datasets (such as weather, solar energy, and electricity consumption) as well as some invented "complex" scenarios.

No One-Size-Fits-All: For some datasets (like simple weather patterns), the "Smart Switch" kept the KAN part mostly closed because a simple calculator was sufficient. For other datasets (like complex regime-switching signals), the switch opened wide, and the KAN part saved the day, reducing errors by nearly 60% compared to simple models.
The Gate Knows Best: The "Smart Switch" automatically determined when the data was too complicated for a simple model and when it was simple enough to ignore the complex part.
Proof of Work: By cutting the wires and seeing the prediction fail, they proved that the model was not just memorizing data; it was actually using specific, complex mathematical shapes to understand the time series.

Summary

This article says: "Do not just look at the pretty graphs inside a neural network. Build a model with a switch that knows when to use complex mathematics, and then prove it works by cutting the wires and seeing if the model collapses."

They showed that their new method is just as accurate as the best existing models, but unlike those, it can actually point to specific time points and specific mathematical shapes that explain why it made a prediction. It turns a magic trick into a transparent, understandable process.

Technical Conclusion: Temporal Functional Circuits in KAN Forecasting

Problem Statement
Long-term time series forecasting (LTSF) has recently demonstrated that simple linear models outperform complex architectures like Transformers on standard benchmarks, raising questions about the necessity of additional model complexity. While Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to Multi-Layer Perceptrons (MLPs) due to their "inherently interpretable" learnable edge functions (typically B-splines), existing applications often rely on visualizing these functions as proof of interpretability. This work argues that visualizing latent spline shapes is not equivalent to providing faithful, temporally anchored explanations. Specifically, a KAN edge function $\phi_e(z)$ does not inherently reveal which input time steps activated it, whether the model actually utilized the edge, or whether its removal would alter the prediction. Furthermore, prior investigations suggest that no single forecasting kernel (linear, KAN, or attention) universally dominates across all datasets, necessitating a framework to determine when KANs are advantageous and why.

Methodology
The work introduces Temporal Functional Circuits, a framework aimed at transforming KAN edge functions into faithful explanations through a combination of architectural design and intervention-based validation of latent visualizations.

Gated Residual KAN Architecture:
The authors propose a forecasting architecture defined as $\hat{y} = f_{lin}(x) + g(x) \odot f_{KAN}(x)$ .
- Decomposition: The input signal is decomposed into trend and residual components using a moving average filter.
- Branches: A linear branch ( $f_{lin}$ ) handles linear patterns, while a KAN branch ( $f_{KAN}$ ) captures nonlinear residuals.
- Gating: An $L_1$ -regularized gate $g(x) \in [0, 1]^C$ controls the contribution of the KAN branch per channel. This enables the model to learn when to suppress the KAN correction (returning to linearity) and provides a measurable metric for KAN usage ( $U_{KAN}$ ).
- KAN Structure: The KAN branch uses splines on the edges, defined as $\phi_e(z) = w \cdot \text{SiLU}(z) + \sum c_k B_p^k(z)$ .
Temporal Functional Circuits (TFC):
For each KAN edge $e$ , the framework constructs an explanation tuple $E_e = (\phi_e, A_e, I_e, \Delta_e)$ :
- $\phi_e$ : The learned univariate function.
- $A_e$ (Lag Assignment): An output-aware assignment method that maps edges to specific input lags by computing gradients through the patch embedding and downstream layers, taking the gate state into account.
- $I_e$ (Importance): A ranking metric based on the expected contribution of the edge to the prediction, combining the edge's output amplitude and downstream sensitivity. The work also utilizes the Activation Range ( $R_e$ ) of the B-spline grid as a data-independent proxy for importance.
- $\Delta_e$ (Interventional Effect): The change in prediction loss upon intervening on the edge. Interventions include zeroing out the edge, replacing it with its mean, or Spline Removal (retaining the base SiLU term while removing the learned B-spline component).
Faithfulness Validation:
The framework validates explanations through:
- Edge Ablation Curves: The stepwise removal of the highest-ranked edges (by $R_e$ ) and measuring MSE degradation. A faithful ranking should show $\Delta_{top-k} > \Delta_{random-k} > \Delta_{bottom-k}$ .
- Recovery of Synthetic Mechanisms: Testing the model on synthetic regimes of increasing complexity (Linear, Multi-Frequency, Threshold AR, Regime Switching) to verify whether gate usage correlates with signal complexity.
- Intervention Tests: Specifically testing whether the learned spline form itself carries predictive value by comparing full zeroing against spline removal.

Main Results

Performance: Across eight benchmarks (including Weather, ETT, Solar, and PPG), the Gated KAN is competitive with linear, attention, and MLP alternatives. It achieves the best or tied-best MSE on 4/8 datasets. Notably, the Gated KAN achieves a 59% lower MSE than purely linear models on regime-switching signals.
Gate Usage: The gate usage metric ( $U_{KAN}$ ) correlates with signal complexity. In synthetic experiments, $U_{KAN}$ increases monotonically from linear (0.49) to regime-switching signals (0.72). On real data, $U_{KAN}$ is low for predominantly linear datasets (e.g., ETTh1) and higher for complex ones (e.g., Solar).
Faithfulness of Explanations:
- Ablation Curves: Removing the highest-ranked edges (by activation range $R_e$ ) consistently degrades performance more than random removal or removing the lowest-ranked edges. On the Weather dataset, removing the top-50 edges increased MSE by $2.20 \times 10^{-4}$ , whereas random removal had a negligible effect.
- Spline Value: Removing only the B-spline component (while retaining the base SiLU) degraded predictions and accounted for up to 41% of the effect of full zeroing on the Weather dataset. This provides evidence that the learned spline form itself carries predictive value beyond the base activation.
- Lag Recovery: On synthetic signals with known causal lags, the proposed edge-to-lag assignment ( $A_e$ ) recovered the true lags with an AUPRC of 0.44 (6x above random) and identified specific KAN edges responsible for lag dependencies.
Comparison with MLP: While a Gated MLP can achieve the accuracy of the Gated KAN, it cannot provide the explicit, verifiable edge functions required for Temporal Functional Circuit analysis.

Significance and Claims
The work claims that KAN interpretability requires more than just visualizing spline plots; it demands temporal anchoring and intervention-tested faithfulness.

Mechanistic Insight: The framework demonstrates that KANs can be decomposed into functional circuits where specific edges are linked to specific input lags and validated through removal.
Diagnostic Utility: The gated architecture provides a measurable diagnosis ( $U_{KAN}$ ) indicating when a nonlinear correction is necessary, explaining why KANs help on some datasets (complex, regime-switching) but not others (predominantly linear).
Value of Splines: The work provides empirical evidence that the learned B-spline form is not merely a fitting artifact but a functional component of the prediction, as its removal significantly impairs performance.
Modest Scope: The authors acknowledge that their interventions establish in-model effects, not causal claims about the data-generating process. They further note that while the Gated KAN is competitive, it is not consistently state-of-the-art, and the framework is a step toward understanding when and how KANs contribute to forecasting, rather than claiming universal superiority.