Gaussian ODE filtering is a probabilistic numerical method to solve ordinary differential equations (ODEs). It computes a Bayesian posterior over the solution from evaluations of the vector field defining the ODE. Its most popular version, which employs an integrated Brownian motion prior, uses Taylor expansions of the mean to extrapolate forward and has the same convergence rates as classical numerical methods. As the solution of many important ODEs are periodic functions (oscillators), we raise the question whether Fourier expansions can also be brought to bear within the framework of Gaussian ODE filtering. To this end, we construct a Fourier state space model for ODEs and a `hybrid' model that combines a Taylor (Brownian motion) and Fourier state space model. We show by experiments how the hybrid model might become useful in cheaply predicting until the end of the time domain.
Manifold-constrained Gaussian process inference for time-varying parameters in dynamic systems
在生物医学研究和其他科学领域中，对动态系统建模时，常微分方程（ODE）中的参数识别是一项重要且具有挑战性的任务，特别是在存在时变参数的情况下。本文提出了一种快速、准确的时变MAnifold-constrained高斯过程推断方法TVMAGI（Time-Varying MAnifold-constrained Gaussian process Inference），利用噪声和稀疏观测数据同时估计常微分方程中的时常数和时变参数。TVMAGI在系统元件的时间序列以及时变参数上施加高斯过程模型，并限制导数过程以满足ODE条件。因此，TVMAGI完全绕过数值积分，并实现了计算时间的大幅节省。通过将ODE结构通过多种约束，TVMAGI享有贝叶斯范式下的原则性统计结构，这进一步使其能够处理系统与缺失数据或未观察到的组件。高斯过程先验也解决了常微分方程中时变参数的可辨识性问题。与现有的方法不同，TVMAGI可以应用于一般的非线性系统，而无需特定的结构假设。三个仿真例子，包括一个传染病房室模型，以说明我们的方法的鲁棒性和效率相比，数值积分和贝叶斯滤波方法。
Nonparametric Filtering, Estimation and Classification using Neural Jump ODEs
Department of Mathematics ETH Zurich（数学系，苏黎世联邦理工学院）；Department of Statistics UC Berkeley（统计系，伯克利大学）；Department of Mathematics Albert-Ludwigs-Universität Freiburg（数学系，弗赖堡阿尔伯特-路易斯大学）
Rotation-Parameterized Graph Fractional Fourier Transform: Definition, Properties, and Optimal Filtering
School of Mathematics and Statistics, Nanjing University of Information Science and Technology（南京信息工程大学数学与统计学院）；School of Communication and Artificial Intelligence, Nanjing Institute of Technology（南京理工大学通信与人工智能学院）；School of Integrated Circuits, Nanjing Institute of Technology（南京理工大学集成电路学院）；Jiangsu Province Engineering Research Center of IntelliSense Technology and System（江苏省智能感知技术与系统工程研究中心）；Hubei Key Laboratory of Applied Mathematics, Hubei University（湖北省应用数学重点实验室）；Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai Jiao Tong University（教育部系统控制与信息处理重点实验室，上海交通大学）
