Recently, Neural Ordinary Differential Equations has emerged as a powerful framework for modeling physical simulations without explicitly defining the ODEs governing the system, but learning them via machine learning. However, the question: Can Bayesian learning frameworks be integrated with Neural ODEs to robustly quantify the uncertainty in the weights of a Neural ODE? remains unanswered. In an effort to address this question, we demonstrate the successful integration of Neural ODEs with two methods of Bayesian Inference: (a) The No-U-Turn MCMC sampler (NUTS) and (b) Stochastic Langevin Gradient Descent (SGLD). We test the performance of our Bayesian Neural ODE approach on classical physical systems, as well as on standard machine learning datasets like MNIST, using GPU acceleration. Finally, considering a simple example, we demonstrate the probabilistic identification of model specification in partially-described dynamical systems using universal ordinary differential equations. Together, this gives a scientific machine learning tool for probabilistic estimation of epistemic uncertainties.

本文讨论了如何通过整合贝叶斯学习框架来量化神经普通微分方程中权重的不确定性，并且展示了在MNIST数据集上使用GPU加速的No-U-Turn MCMC采样器、Stochastic Gradient Hamiltonian Monte Carlo和Stochastic Langevin Gradient Descent等推理方法成功集成神经ODE的实验结果。然后，我们首次证明了变分推理与标准化流和神经ODE的成功整合，生成了强大的贝叶斯神经ODE对象。最后，我们演示了如何利用普适的常微分方程概率地识别部分描述的动力系统中的模型规范，从而为探索认识上的不确定性提供了科学的机器学习工具。

贝叶斯神经常微分方程