将拉盖尔基与巴伦函数结合，实现带不确定性的因果系统建模。
Barron-Wiener-Laguerre models
- 用巴伦函数视角重构非线性部分，支持贝叶斯推断
- 在经典拉盖尔基线性动态基础上加入不确定性量化
- 适合需要可信预测的时序建模与系统识别任务
我们提出一种概率化的因果算子学习方法，扩展了经典的维纳-拉盖尔模型。传统模型使用正交拉盖尔基表示稳定线性动态，并通过静态非线性映射处理特征，结构简洁但仅提供确定性点估计。本文从巴伦函数逼近角度重新审视非线性部分，将两层网络、随机傅里叶特征和极限学习机视为参数测度积分表示的离散化形式。这一视角自然支持对非线性映射的贝叶斯推断，从而获得后验预测不确定性。通过结合拉盖尔参数化的因果动态与概率型巴伦式非线性逼近器，我们构建了一类结构清晰且表达能力强的因果算子，具备不确定性量化能力。该框架融合经典系统辨识与现代测度函数逼近思想，为时序建模与非线性系统辨识提供了严谨方法。
原文摘要 · Abstract (English)
We propose a probabilistic extension of Wiener-Laguerre models for causal operator learning. Classical Wiener-Laguerre models parameterize stable linear dynamics using orthonormal Laguerre bases and apply a static nonlinear map to the resulting features. While structurally efficient and interpretable, they provide only deterministic point estimates. We reinterpret the nonlinear component through the lens of Barron function approximation, viewing two-layer networks, random Fourier features, and extreme learning machines as discretizations of integral representations over parameter measures. This perspective naturally admits Bayesian inference on the nonlinear map and yields posterior predictive uncertainty. By combining Laguerre-parameterized causal dynamics with probabilistic Barron-type nonlinear approximators, we obtain a structured yet expressive class of causal operators equipped with uncertainty quantification. The resulting framework bridges classical system identification and modern measure-based function approximation, providing a principled approach to time-series modeling and nonlinear systems identification.
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