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

2026-05-07

AI Summary

Time series forecasting often lacks interpretability, particularly in complex dynamical systems where tracing the rationale behind predictions is challenging. This work proposes a temporal functional circuit framework based on gated residual Kolmogorov–Arnold Networks (KANs), decomposing forecasts into a linear basis plus a sparse KAN correction term. Verifiable explanations are achieved through output-aware attribution, activation range ranking, and edge-level interventions—such as zeroing out or spline removal. For the first time, KAN’s edge functions are transformed into a reliable mechanism for temporal interpretability, revealing that B-spline shapes themselves carry predictive value independent of activation functions. Experiments show that the gating mechanism adaptively activates with signal complexity, reduces MSE by 59% over purely linear models on regime-switching signals, and matches the performance of linear, attention-based, and MLP models across eight benchmarks while providing interpretable edge functions inaccessible to those alternatives.

Abstract

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.

Problem

• Kolmogorov-Arnold Networks
• time-series forecasting
• interpretability
• temporal explanations
• functional circuits

Innovation

• Temporal Functional Circuits
• Kolmogorov-Arnold Networks
• faithful explanation
• gated residual KAN
• edge-level intervention