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

arXiv: https://arxiv.org/abs/2605.05685 · HTML: https://mlsanity.com/paper/2605.05685/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 41 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 27 of 27)

- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [Sanity Checks for Saliency Maps](https://mlsanity.com/paper/1810.03292/) (2018)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Attention is not Explanation](https://mlsanity.com/paper/1902.10186/) (2019)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Attention is not not Explanation](https://mlsanity.com/paper/1908.04626/) (2019)
- [Towards Faithfully Interpretable NLP Systems: How should we define and evaluate faithfulness?](https://mlsanity.com/paper/2004.03685/) (2020)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Benchmarking Deep Learning Interpretability in Time Series Predictions](https://mlsanity.com/paper/2010.13924/) (2020)
- [Kolmogorov-Arnold Networks (KANs) for Time Series Analysis](https://mlsanity.com/paper/2405.08790/) (2024)
- [Explaining Time Series Predictions with Dynamic Masks](https://mlsanity.com/paper/2106.05303/) (2021)
- [Encoding Time-Series Explanations through Self-Supervised Model Behavior Consistency](https://mlsanity.com/paper/2306.02109/) (2023)
- [TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting](https://mlsanity.com/paper/2502.06910/) (2025)
- [Mixture-of-Linear-Experts for Long-term Time Series Forecasting](https://mlsanity.com/paper/2312.06786/) (2023)
- [Temporal Dependencies in Feature Importance for Time Series Predictions](https://mlsanity.com/paper/2107.14317/) (2021)
- [A Gated Residual Kolmogorov-Arnold Networks for Mixtures of Experts](https://mlsanity.com/paper/2409.15161/) (2024)
- [CausalKANs: interpretable treatment effect estimation with Kolmogorov-Arnold networks](https://mlsanity.com/paper/2509.22467/) (2025)
- [QuantKAN: A Unified Quantization Framework for Kolmogorov Arnold Networks](https://mlsanity.com/paper/2511.18689/) (2025)
- [Time series forecasting with Hahn Kolmogorov-Arnold networks](https://mlsanity.com/paper/2601.18837/) (2026)
- [DecompKAN: Decomposed Patch-KAN for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2604.23968/) (2026)

## Abstract (arXiv)

Unlike MLPs, Kolmogorov-Arnold Networks (KANs) expose explicit learnable edge functions on every connection, enabling mechanistic explanation in time-series forecasting. This paper introduces Temporal Functional Circuits, a framework that transforms KAN edge functions from latent visualizations into faithful, temporally grounded explanations. Built on a gated residual KAN that decomposes forecasts into a linear base and a sparsely activated KAN correction, the framework (i) maps each edge to input lags via output-aware attribution, (ii) ranks edges by learned activation range, and (iii) validates faithfulness through edge-level interventions including zeroing and spline removal. Removing the learned B-spline component while retaining the base SiLU term degrades forecasts, providing evidence that the spline shape itself carries predictive value beyond the base activation. On four synthetic regimes of increasing complexity, the learned gate opens progressively wider as signal complexity grows. On regime-switching signals, gated KAN achieves 59% lower MSE than linear-only models. Across eight benchmarks, the gated architecture is competitive with linear, attention, and MLP alternatives, while providing interpretable edge functions that MLP-based corrections cannot offer.