We identify reduced order models (ROM) of forced systems from data using invariant foliations. The forcing can be external, parametric, periodic or quasi-periodic. The process has four steps: 1. identify an approximate invariant torus and the linear dynamics about the torus; 2. identify a globally defined invariant foliation about the torus; 3. identify a local foliation about an invariant manifold that complements the global foliation 4. extract the invariant manifold as the leaf going through the torus and interpret the result. We combine steps 2 and 3, so that we can track the location of the invariant torus and scale the invariance equations appropriately. We highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.

翻译：我们利用不变叶状结构从数据中识别受迫系统的降阶模型(ROM)。受迫可以是外部的、参数的、周期性的或准周期性的。该过程包含四个步骤：1. 识别近似不变环面及其线性动力学；2. 识别环面附近全局定义的不变叶状结构；3. 识别补充全局叶状结构的不变流形局部叶状结构；4. 提取通过环面的叶片作为不变流形并解释结果。我们将步骤2和3合并，从而能够追踪不变环面的位置并适当缩放不变性方程。我们指出了不变流形和叶状结构在数据拟合中存在的若干基本限制，这需要进一步的数学方法来解决。