Bayesian Neural Ordinary Differential Equations

Abstract

Recently, Neural Ordinary Differential Equations has emerged as a powerful framework for modeling physical simulations without explicitly defining the ODEs governing the system, but instead learning them via machine learning. However, the question: "Can Bayesian learning frameworks be integrated with Neural ODE's to robustly quantify the uncertainty in the weights of a Neural ODE?" remains unanswered. In an effort to address this question, we primarily evaluate the following categories of inference methods: (a) The No-U-Turn MCMC sampler (NUTS), (b) Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) and (c) Stochastic Langevin Gradient Descent (SGLD). We demonstrate the successful integration of Neural ODEs with the above Bayesian inference frameworks on classical physical systems, as well as on standard machine learning datasets like MNIST, using GPU acceleration. On the MNIST dataset, we achieve a posterior sample accuracy of 98.5% on the test ensemble of 10,000 images. Subsequently, for the first time, we demonstrate the successful integration of variational inference with normalizing flows and Neural ODEs, leading to a powerful Bayesian Neural ODE object. Finally, considering a predator-prey model and an epidemiological system, 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.

16 pages, 10 figures, 3 tables; added new inference methods, substantially improved MNIST accuracy, revised author affiliations

References in corpus (7)

• Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users
• Learning in Implicit Generative Models
• A Comprehensive guide to Bayesian Convolutional Neural Network with Variational Inference
• Subspace Inference for Bayesian Deep Learning
• Learning Model Reparametrizations: Implicit Variational Inference by Fitting MCMC distributions
• Differential Bayesian Neural Nets
• Bayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Data

Cited by in corpus (5)

• Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations
• Learning orbital dynamics of binary black hole systems from gravitational wave measurements
• Arrhenius.jl: A Differentiable Combustion SimulationPackage
• Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunities
• Analysis of ODE2VAE with Examples