论文标题
通过与约束的高斯过程回归来推断微分方程中未知参数
Inferring the unknown parameters in Differential Equation by Gaussian Process Regression with Constraint
论文作者
论文摘要
微分方程（DE）是在金融和生物学等各种科学主题中常用的建模方法。 DE模型中的参数通常具有有趣的科学解释，但是它们的价值通常是未知的，需要从DE的测量值中估算。在这项工作中，我们提出了一个贝叶斯推理框架，以解决仅给定噪声和稀缺的解决方案的估计模型参数的问题。这个问题的一个关键问题是，在以差分方程的形式假设函数及其导数的形式的假设下，从给定位置点的函数值嘈杂地估算功能的衍生物。为了解决关键问题，我们使用使用约束方法（GPRC）方法的高斯过程回归，该方法将解决方案，衍生物和参数微分方程共同建模，以估计解决方案及其衍生物。对于非线性微分方程，使用了线性化方法的Picard-tieration样近似，以便GPRC仍然可以迭代地适用。将数据和方程信息结合在一起的新潜力是在我们推论的可能性中提出的。通过数值示例，我们说明所提出的方法具有竞争性能，以估计DES中未知参数的现有方法。
Differential Equation (DE) is a commonly used modeling method in various scientific subjects such as finance and biology. The parameters in DE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the DE. In this work, we propose a Bayesian inference framework to solve the problem of estimating the parameters of the DE model, from the given noisy and scarce observations of the solution only. A key issue in this problem is to robustly estimate the derivatives of a function from noisy observations of only the function values at given location points, under the assumption of a physical model in the form of differential equation governing the function and its derivatives. To address the key issue, we use the Gaussian Process Regression with Constraint (GPRC) method which jointly model the solution, the derivatives, and the parametric differential equation, to estimate the solution and its derivatives. For nonlinear differential equations, a Picard-iteration-like approximation of linearization method is used so that the GPRC can be still iteratively applicable. A new potential which combines the data and equation information, is proposed and used in the likelihood for our inference. With numerical examples, we illustrate that the proposed method has competitive performance against existing approaches for estimating the unknown parameters in DEs.
