Recently, Gaussian processes have been utilized to model the vector field of continuous dynamical systems. Bayesian inference for such models \cite{hegde2022variational} has been extensively studied and has been applied in tasks such as time series prediction, providing uncertain estimates. However, previous Gaussian Process Ordinary Differential Equation (ODE) models may underperform on datasets with non-Gaussian process priors, as their constrained priors and mean-field posteriors may lack flexibility. To address this limitation, we incorporate normalizing flows to reparameterize the vector field of ODEs, resulting in a more flexible and expressive prior distribution. Additionally, due to the analytically tractable probability density functions of normalizing flows, we apply them to the posterior inference of GP ODEs, generating a non-Gaussian posterior. Through these dual applications of normalizing flows, our model improves accuracy and uncertainty estimates for Bayesian Gaussian Process ODEs. The effectiveness of our approach is demonstrated on simulated dynamical systems and real-world human motion data, including tasks such as time series prediction and missing data recovery. Experimental results indicate that our proposed method effectively captures model uncertainty while improving accuracy.

翻译：近期，高斯过程被用于建模连续动力系统的向量场。针对此类模型的贝叶斯推理方法已被广泛研究，并应用于时间序列预测等任务中，能够提供不确定性估计。然而，传统的高斯过程常微分方程（ODE）模型在处理非高斯过程先验的数据集时可能表现不佳，因为其受限于约束性先验和平均场后验的灵活性不足。为解决这一局限，我们引入归一化流对常微分方程的向量场进行重参数化，从而获得更灵活且表现力更强的先验分布。此外，利用归一化流具有解析可追踪概率密度函数的特性，我们将其应用于高斯过程常微分方程的后验推理，生成非高斯后验分布。通过归一化流的双重应用，我们的模型提升了贝叶斯高斯过程常微分方程的精度与不确定性估计能力。在模拟动力系统与真实人体运动数据（包括时间序列预测与缺失数据恢复任务）上的实验表明，所提方法在提升准确性的同时，能够有效捕捉模型不确定性。