从混沌中提取信号：用于动力系统贝叶斯分析的MAGI方法进展
Extracting Signal out of Chaos: Advancements on MAGI for Bayesian Analysis of Dynamical Systems
AI总结：
本文提出pMAGI和PMSP两种新方法，改进MAGI框架以在稀疏噪声数据下实现ODE动力系统的贝叶斯参数推断、轨迹重建、稳定性分类与多步预测，并显著优于PINN方法。
摘要：
本文在基于流形约束的高斯过程推断（MAGI）方法的基础上展开研究，该方法用于基于常微分方程（ODE）的动力系统的贝叶斯参数推断和轨迹重建，重点关注稀疏和噪声数据条件。首先，我们引入了Pilot MAGI（pMAGI），这是对基础MAGI方法的一种新颖方法学升级，显著提高了数值稳定性、参数推断和轨迹重建性能。其次，据我们所知，我们首次展示了如何将基于MAGI的方法与动力系统理论相结合，对系统是稳定还是混沌提供概率分类。第三，我们证明了pMAGI在许多场景下相对于计算成本高得多且过度参数化的方法表现出色。第四，我们引入了Pilot MAGI序列预测（PMSP），这是一种基于pMAGI的新方法，允许在仅有稀疏和噪声观测的情况下，预测基于ODE的动力系统未来多个时间步的轨迹。我们表明，PMSP即使在混沌动力系统上也能输出准确的未来预测，并显著优于基于PINN的方法。总体而言，我们为文献贡献了两种新方法，即pMAGI和PMSP，它们作为物理信息神经网络（Physics-Informed Neural Network）的贝叶斯、不确定性量化的替代方案。
英文摘要：
This work builds off the manifold-constrained Gaussian process inference (MAGI) method for Bayesian parameter inference and trajectory reconstruction of ODE-based dynamical systems, focusing primarily on sparse and noisy data conditions. First, we introduce Pilot MAGI (pMAGI), a novel methodological upgrade on the base MAGI method that confers significantly-improved numerical stability, parameter inference, and trajectory reconstruction. Second, we demonstrate, for the first time to our knowledge, how one can combine MAGI-based methods with dynamical systems theory to provide probabilistic classifications of whether a system is stable or chaotic. Third, we demonstrate how pMAGI performs favorably in many settings against much more computationally-expensive and overparameterized methods. Fourth, we introduce Pilot MAGI Sequential Prediction (PMSP), a novel method building upon pMAGI that allows one to predict the trajectory of ODE-based dynamical systems multiple time steps into the future, given only sparse and noisy observations. We show that PMSP can output accurate future predictions even on chaotic dynamical systems and significantly outperform PINN-based methods. Overall, we contribute to the literature two novel methods, pMAGI and PMSP, that serve as Bayesian, uncertainty-quantified competitors to the Physics-Informed Neural Network.
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