论文标题
通过高斯流程回归对数据和微分方程的衍生物的明确估计
Explicit Estimation of Derivatives from Data and Differential Equations by Gaussian Process Regression
论文作者
论文摘要
在这项工作中，我们采用贝叶斯推理框架来解决估计解决方案的问题，尤其是它的衍生物（从给定的嘈杂和稀缺观察到解决方案数据）中满足已知微分方程的衍生物。为了解决衍生估计的准确性和鲁棒性的关键问题，我们使用高斯工艺共同对解决方案，衍生物和微分方程进行建模。通过将线性微分方程作为线性约束，开发了具有约束方法（GPRC）的高斯过程回归，以提高衍生物预测的准确性。对于非线性微分方程，我们提出了仅从数据获得的高斯过程周围的PICARD-类似线性化的近似，以便我们的GPRC仍然可以迭代地适用。此外，还采用专家方法的产物来确保认为初始或边界条件以进一步提高衍生物的预测准确性。我们提出了几个数值结果，以说明与标准数据驱动的高斯过程回归相比，我们的新方法的优势。
In this work, we employ the Bayesian inference framework to solve the problem of estimating the solution and particularly, its derivatives, which satisfy a known differential equation, from the given noisy and scarce observations of the solution data only. To address the key issue of accuracy and robustness of derivative estimation, we use the Gaussian processes to jointly model the solution, the derivatives, and the differential equation. By regarding the linear differential equation as a linear constraint, a Gaussian process regression with constraint method (GPRC) is developed to improve the accuracy of prediction of derivatives. For nonlinear differential equations, we propose a Picard-iteration-like approximation of linearization around the Gaussian process obtained only from data so that our GPRC can be still iteratively applicable. Besides, a product of experts method is applied to ensure the initial or boundary condition is considered to further enhance the prediction accuracy of the derivatives. We present several numerical results to illustrate the advantages of our new method in comparison to the standard data-driven Gaussian process regression.
