Learning a nonparametric system of ordinary differential equations (ODEs) from $n$ trajectory snapshots in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach to learning using the implicit formulation provided by vector-valued Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral form, which we subsequently minimize, we derive our learning algorithm. The minimization problem's solution for the vector field relies on multivariate occupation kernel functions associated with the solution trajectories. We validate our approach through experiments on highly nonlinear simulated and real data, where $d$ may exceed 100. We further demonstrate the versatility of the proposed method by learning a nonparametric first order quasilinear partial differential equation.

翻译：从$d$维状态空间中的$n$条轨迹快照学习非参数常微分方程组，需要学习$d$个关于$d$变量的函数。除非系统属性（如稀疏性和对称性）已知，显式公式的规模与$d$呈二次方增长。本文提出一种线性学习方法，利用向量值再生核希尔伯特空间提供的隐式公式。通过将常微分方程重写为更弱的积分形式并随后最小化，我们推导出学习算法。向量场的极小化问题解依赖于与解轨迹相关的多元占用核函数。我们通过在高度非线性的模拟数据和真实数据上的实验验证了该方法，其中$d$可超过100。此外，我们通过学习非参数一阶拟线性偏微分方程进一步展示了所提方法的通用性。