自适应非线性向量自回归模型提升噪声混沌系统预测精度
Adaptive Nonlinear Vector Autoregression: Robust Forecasting for Noisy Chaotic Time Series
- 用可训练MLP生成数据驱动的非线性特征，联合优化线性读出层
- 在无噪和合成噪声下，对洛伦兹-63等系统预测误差更低
- 适合高噪声、高维混沌系统建模，尤其适用于实际观测数据
非线性向量自回归（NVAR）和储备池计算（RC）在预测混沌动力系统（如洛伦兹-63模型和厄尔尼诺-南方涛动）方面表现良好。然而，它们依赖固定的非线性变换——NVAR使用多项式展开，RC使用随机特征映射——限制了其在高噪声或复杂真实数据中的适应性。此外，这些方法在高维场景中因优化过程中的矩阵求逆代价高昂而表现出较差的可扩展性。本文提出一种数据自适应的NVAR模型，将延迟嵌入的线性输入与浅层可训练多层感知机（MLP）生成的特征结合。与标准NVAR和RC不同，该模型通过基于梯度的优化联合训练MLP与线性读出层，使模型能学习数据驱动的非线性关系，同时保持简单读出结构并提升可扩展性。在多个混沌系统上进行的初步实验，涵盖无噪和合成噪声条件，表明该自适应模型在预测精度上优于标准NVAR、最常用的RC模型漏失回声状态网络（ESN）以及混合型ESN，展现出在噪声条件下的鲁棒预测能力。
原文摘要 · Abstract (English)
Nonlinear vector autoregression (NVAR) and reservoir computing (RC) have shown promise in forecasting chaotic dynamical systems, such as the Lorenz-63 model and El Nino-Southern Oscillation. However, their reliance on fixed nonlinear transformations - polynomial expansions in NVAR or random feature maps in RC - limits their adaptability to high noise or complex real-world data. Furthermore, these methods also exhibit poor scalability in high-dimensional settings due to costly matrix inversion during optimization. We propose a data-adaptive NVAR model that combines delay-embedded linear inputs with features generated by a shallow, trainable multilayer perceptron (MLP). Unlike standard NVAR and RC models, the MLP and linear readout are jointly trained using gradient-based optimization, enabling the model to learn data-driven nonlinearities, while preserving a simple readout structure and improving scalability. Initial experiments across multiple chaotic systems, tested under noise-free and synthetically noisy conditions, showed that the adaptive model outperformed in predictive accuracy the standard NVAR, a leaky echo state network (ESN) - the most common RC model - and a hybrid ESN, thereby showing robust forecasting under noisy conditions.
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