Lahouel Kamel, Wells Michael, Rielly Victor, Lew Ethan, Lovitza David, Jedynak Bruno M
TGen, 445 N. Fifth Street, Phoenix, AZ 85004.
Dept. of Math & Stat, Portland State University, 1855 SW Broadway, Portland, OR 97201.
J Comput Phys. 2024 Jun 15;507. doi: 10.1016/j.jcp.2024.112971. Epub 2024 Mar 29.
Learning nonparametric systems of Ordinary Differential Equations (ODEs) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for for which the solution of the ODE exists and is unique. Learning consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the distance between and its estimator. Experiments are provided for the FitzHugh-Nagumo oscillator, the Lorenz system, and for predicting the Amyloid level in the cortex of aging subjects. In all cases, we show competitive results compared with the state-of-the-art.
从噪声数据中学习常微分方程（ODE）的非参数系统是一个新兴的机器学习主题。我们使用成熟的再生核希尔伯特空间（RKHS）理论来定义ODE解存在且唯一的候选对象。学习过程包括在RKHS中解决一个约束优化问题。我们提出一种惩罚方法，该方法迭代地使用表示定理和欧拉近似来提供数值解。我们证明了真实ODE与其估计值之间距离的泛化界。针对菲茨休 - 纳古莫振荡器、洛伦兹系统以及预测衰老受试者皮质中的淀粉样蛋白水平进行了实验。在所有情况下，我们都展示了与当前最优方法相比具有竞争力的结果。
