D-CODE: Discovering Closed-form ODEs from Observed Trajectories
D-CODE: Discovering Closed-form ODEs from Observed Trajectories
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发表时间：
2022
影响因子：
2.4
通讯作者：
Zhaozhi Qian;Krzysztof Kacprzyk;M. Schaar
中科院分区：
文献类型：
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作者：
Zhaozhi Qian;Krzysztof Kacprzyk;M. Schaar
几个世纪以来，科学家们一直在手动设计封闭形式的常微分方程式(ODE)来模拟动力系统。从观察到的轨迹中提取闭合形式的ODE的自动化工具将加快建模过程。传统上，符号回归被用来发现以标签-特征对(ai，bi)作为训练样本的闭式预测函数a=f(B)。然而，常微分方程组对动力系统的时间导数ẋ(T)进行建模，例如ẋ(T)=f(x(T)，t)，并且通常不能观察到“标签”ẋ(T)。现有的弥补这一差距的方法仅适用于具有低测量噪声和频繁采样的窄范围设置。在这项工作中，我们提出了封闭形式的ODE发现框架(D-CODE)，它使符号回归超越了有监督学习的范式。D-CODE使用一种基于常微分方程组变分公式的新目标函数来绕过不可观测的时间导数。对于形式证明，我们证明了该目标是真实(但未知的)ODE估计误差的有效代理。在实验中，D-CODE成功地发现了在具有高噪声和不频繁采样的挑战性测量设置下的各种动力系统的控制方程。
For centuries, scientists have manually designed closed-form ordinary differential equations (ODEs) to model dynamical systems. An automated tool to distill closedform ODEs from observed trajectories would accelerate the modeling process. Traditionally, symbolic regression is used to uncover a closed-form prediction function a = f(b) with label-feature pairs (ai, bi) as training examples. However, an ODE models the time derivative ẋ(t) of a dynamical system, e.g. ẋ(t) = f(x(t), t), and the “label” ẋ(t) is usually not observed. The existing ways to bridge this gap only perform well for a narrow range of settings with low measurement noise and frequent sampling. In this work, we propose the Discovery of Closedform ODE framework (D-CODE), which advances symbolic regression beyond the paradigm of supervised learning. D-CODE uses a novel objective function based on the variational formulation of ODEs to bypass the unobserved time derivative. For formal justification, we prove that this objective is a valid proxy for the estimation error of the true (but unknown) ODE. In the experiments, D-CODE successfully discovered the governing equations of a diverse range of dynamical systems under challenging measurement settings with high noise and infrequent sampling.