用弱形式核岭回归提升噪声下动力系统学习精度
Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression
- 将弱形式与核岭回归结合，实现噪声数据过滤
- 在64维混沌系统和1.5万维流体数据上表现更优
- 适合科研人员处理高维噪声动力学问题
从噪声测量中准确预测复杂动力系统仍是科学计算中的重大挑战。核岭回归在干净数据上表现良好，但在噪声数据上效果有限。近期研究表明，弱形式具有滤波噪声的能力，不同学习策略在弱形式框架下已实现更强的抗噪性。本文揭示了弱形式背后的滤波机制，并提供偏差-方差误差分解。基于此，我们将弱形式与核学习策略结合，提出弱形式核岭回归（WKRR）用于学习动力系统。该框架简单易实现，对干净与噪声数据均有效，性能优于多个基线方法。我们在高达64维的混沌基准系统以及15,000维的真实流体数据上验证了WKRR的性能。
原文摘要 · Abstract (English)
Accurate prediction of complex dynamical systems from noisy measurements remains a significant challenge in scientific computing. Kernel ridge regression learning strategies are often effective when applied to clean data, but have limited success with noisy data. Recent work has observed that a weak formulation can act to filter noisy data, and different learning strategies have achieved increased noise robustness with a weak-form framework. In this manuscript, we give an overview of the filtering mechanism behind the weak formulation and provide a bias-variance error decomposition. Using these insights, we combine a weak formulation with a kernel learning strategy to propose Weak-form Kernel Ridge Regression (WKRR) for learning dynamical systems. The proposed framework is simple to implement, effective for both clean and noisy data, and outperforms several baseline methods. We demonstrate the performance of WKRR on chaotic benchmark systems in up to 64 dimensions, as well as 15,000-dimensional real-world fluid data.
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