具有边信息的动力系统的学习
Learning Dynamical Systems with Side Information
摘要：
我们提出了一个数学和计算框架的问题，学习动力系统从嘈杂的观察几个轨迹和边信息。边信息是除了轨迹数据之外，我们可能拥有的关于我们想要学习的动力系统的任何知识。它通常是从特定领域的知识或科学学科的基本原则推断出来的。我们感兴趣的是明确地将边信息集成到学习过程中，以弥补轨迹观测的不足。我们确定了六种类型的边信息，自然出现在许多应用中，并导致凸约束的学习问题。首先，我们表明，当我们的模型为未知的动力系统参数化为一个多项式，可以施加我们的边信息约束计算通过半定规划。然后，我们展示了边信息的附加值，用于学习物理学和细胞生物学中基本模型的动态，以及用于学习和控制流行病学中模型的动态。最后，我们研究了多项式动力系统如何在满足边信息（精确或近似）的情况下逼近连续可微系统。我们的整体学习方法结合了凸优化，实代数，动力系统和功能逼近论的想法，并可能导致这些领域之间的新的协同作用。
英文摘要：
We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or basic principles of a scientific discipline. We are interested in explicitly integrating side information into the learning process in order to compensate for scarcity of trajectory observations. We identify six types of side information that arise naturally in many applications and lead to convex constraints in the learning problem. First, we show that when our model for the unknown dynamical system is parameterized as a polynomial, one can impose our side information constraints computationally via semidefinite programming. We then demonstrate the added value of side information for learning the dynamics of basic models in physics and cell biology, as well as for learning and controlling the dynamics of a model in epidemiology. Finally, we study how well polynomial dynamical systems can approximate continuously-differentiable ones while satisfying side information (either exactly or approximately). Our overall learning methodology combines ideas from convex optimization, real algebra, dynamical systems, and functional approximation theory, and can potentially lead to new synergies between these areas.
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