A deterministic temporal process can be determined by its trajectory, an element in the product space of (a) initial condition $z_0 \in \mathcal{Z}$ and (b) transition function $f: (\mathcal{Z}, \mathcal{T}) \to \mathcal{Z}$ often influenced by the control of the underlying dynamical system. Existing methods often model the transition function as a differential equation or as a recurrent neural network. Despite their effectiveness in predicting future measurements, few results have successfully established a method for sampling and statistical inference of trajectories using neural networks, partially due to constraints in the parameterization. In this work, we introduce a mechanism to learn the distribution of trajectories by parameterizing the transition function $f$ explicitly as an element in a function space. Our framework allows efficient synthesis of novel trajectories, while also directly providing a convenient tool for inference, i.e., uncertainty estimation, likelihood evaluations and out of distribution detection for abnormal trajectories. These capabilities can have implications for various downstream tasks, e.g., simulation and evaluation for reinforcement learning.

翻译：确定性时序过程可由其轨迹完全确定，该轨迹定义在下列两个要素的乘积空间中：(a)初始条件 $z_0 \in \mathcal{Z}$ 和 (b)经常受底层动力学系统控制影响的转移函数 $f: (\mathcal{Z}, \mathcal{T}) \to \mathcal{Z}$ 。现有方法通常将转移函数建模为微分方程或递归神经网络。尽管这些方法在预测未来观测值方面表现有效，但鲜有研究成功建立基于神经网络的轨迹采样与统计推断方法，部分原因在于参数化过程中的约束限制。本研究提出一种通过学习轨迹分布的新机制，通过将转移函数 $f$ 显式参数化为函数空间中的元素来实现。该框架既能高效生成新型轨迹，又能直接提供便捷的推断工具（即异常轨迹的不确定性估计、似然评估与分布外检测）。这些能力可对多项下游任务产生重要影响，例如强化学习中的仿真与评估。