论文标题
学习Koopman运营商的代表定理
Representer Theorem for Learning Koopman Operators
论文作者
论文摘要
在这项工作中，考虑了离散时间自主系统的Koopman操作员的问题。学习问题被提出为线性操作员无限维空间中受约束的正规化经验损失最小化。我们表明，在某些但一般的条件下，代表定理涉及学习问题。这允许在有限维空间中重新解决该问题，而无需任何近似和丧失精度。之后，我们考虑了学习问题中的各种正则化和约束案例，包括操作员规范，弗罗贝尼乌斯规范，等级，核规范和稳定性。随后，我们得出相应的有限维问题。此外，我们讨论了所提出的公式与扩展动态模式分解之间的联系。最后，我们提供了一个说明性的数值示例。
In this work, the problem of learning Koopman operator of a discrete-time autonomous system is considered. The learning problem is formulated as a constrained regularized empirical loss minimization in the infinite-dimensional space of linear operators. We show that under certain but general conditions, a representer theorem holds for the learning problem. This allows reformulating the problem in a finite-dimensional space without any approximation and loss of precision. Following this, we consider various cases of regularization and constraints in the learning problem, including the operator norm, the Frobenius norm, rank, nuclear norm, and stability. Subsequently, we derive the corresponding finite-dimensional problem. Furthermore, we discuss the connection between the proposed formulation and the extended dynamic mode decomposition. Finally, we provide an illustrative numerical example.
