Most physical processes posses structural properties such as constant energies, volumes, and other invariants over time. When learning models of such dynamical systems, it is critical to respect these invariants to ensure accurate predictions and physically meaningful behavior. Strikingly, state-of-the-art methods in Gaussian process (GP) dynamics model learning are not addressing this issue. On the other hand, classical numerical integrators are specifically designed to preserve these crucial properties through time. We propose to combine the advantages of GPs as function approximators with structure preserving numerical integrators for dynamical systems, such as Runge-Kutta methods. These integrators assume access to the ground truth dynamics and require evaluations of intermediate and future time steps that are unknown in a learning-based scenario. This makes direct inference of the GP dynamics, with embedded numerical scheme, intractable. Our key technical contribution is the evaluation of the implicitly defined Runge-Kutta transition probability. In a nutshell, we introduce an implicit layer for GP regression, which is embedded into a variational inference-based model learning scheme.

翻译：多数物理过程都拥有结构属性, 如恒定能量、 体积和其他变异性 。 当学习这些动态系统的模型时, 必须尊重这些变量以确保准确的预测和具有物理意义的行为。 令人惊讶的是, 高盛进程动态模型学习中最先进的方法并没有解决这个问题。 另一方面, 古典数字集成器是专门设计来保存这些关键属性的。 我们提议将GPs作为功能匹配器的优势与保持动态系统数字集成器的结构( 如 Runge- Kutta 方法) 结合起来。 这些整合器将利用地面的真相动态, 并要求对学习假设中未知的中间和未来的时间步骤进行评估。 这直接推断GP动态, 并带有嵌入的数值图案, 难以理解。 我们的主要技术贡献是评估隐性定义的 Runge- Kutta 过渡概率。 在坚果中, 我们引入了GP回归的隐含层, 它将嵌入基于变形推断模型学习计划。