回归类问题中相关特征子集与子空间的联合同步学习

中文摘要 AI 辅助

我们对新近提出的熵最优流形聚类（Entropy-Optimal Manifold Clustering, EOMC）进行扩展，使其能够在非平稳和非线性回归问题中联合同步识别相关特征的子集与子空间。结果表明，该扩展方法——我们将其命名为熵最优流形回归（Entropy-Optimal Manifold Regression, EOMR）——可实现鲁棒学习，且迭代复杂度与内存复杂度呈线性缩放。我们将EOMR与作者可获取的人工智能（Artificial Intelligence, AI）和机器学习（Machine Learning, ML）领域最完整的一组最新工具，在混沌与流体动力学的极具挑战性问题上进行对比：（i）在强混沌与极强混沌状态下预测Lorenz-96系统的动力学（强迫参数分别为F=8和F=12）；（ii）在托卡马克等离子体边缘的长谷川-若田模型（Hasegawa-Wakatani model）数据上进行验证。研究表明，上述基准问题（i）和（ii）对当前最先进的ML和AI工具而言确实极具挑战性：因为通用梯度提升随机森林、深度神经网络以及适用于高维小数据学习的基于Transformer的AI工具TabPFN v.03，与EOMR相比，其均方根预测误差相差数个数量级，模型复杂度也高出数个数量级。对于长谷川-若田模型示例，EOMR提炼出了主导本征正交函数（Essential Orthogonal Function, EOF）动力学的极为简单且有效的描述，该描述由仅含8个参数的线性、因果且弱平稳自回归过程构成。

英文摘要

We extend a recently introduced Entropy-Optimal Manifold Clustering (EOMC) to allow for a joint simultaneous identification of subsets and subspaces of relevant features in nonstationary and nonlinear regression problems. It is shown that the proposed extension - that we coin as Entropy-Optimal Manifold Regression (EOMR) - allows a robust learning with linearly-scaling iteration and memory complexities. EOMR is compared to the most complete set of state-of-the-art tools from the Artificial Intelligence (AI) and Machine Learning (ML) that is available to the author, on the very challenging problems from chaotic and fluid dynamics: (i) on predicting the Lorenz-96 systems dynamics in strongly- and very-strongly chaotic regimes (with forcing parameter being $F=8$ and $F=12$, respectively); and, (ii) on a data from the Hasegawa-Wakatani model on the edge of the tokamak plasma. It is demonstrated that the proposed benchmarks (i) and (ii), indeed, are the very challenging problems for the state of the art ML and AI tools - since both the general-purpose gradient boosted random forests and deep neuronal networks, as well as transformer-based AI tools like TabPFN v.03 (more spezialised for large-dimensional small data learning problems) - result in orders of magnitude inferior root mean squared prediction errors, and orders of magnitude larger model complexities, when compared to the EOMR. For a Hasegawa-Wakatani example, EOMR distills a very simple entropy-optimal and skilful description of the leading Essential Orthogonal Function (EOF) dynamics, given by linear, causal and weakly-stationary autoregressive process described by just 8 parameters.

发表机构

• RPTU Kaiserslautern-Landau（凯撒斯劳滕-兰道莱茵-普法尔茨理工大学）

机构由 AI 辅助整理，请以论文原文为准。