Reproduce “Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning” on OpenResearch

About this paper

Zongren Zou, Tingwei Meng, Paula Chen, Jérôme Darbon, George Em Karniadakis

This research establishes a new theoretical connection between Bayesian uncertainty quantification in scientific machine learning and viscous Hamilton-Jacobi partial differential equations. The work then develops a computationally efficient Riccati-based methodology for linear models with Gaussian assumptions, enabling capabilities like continual learning, continuous hyperparameter tuning, and efficient data streaming without retraining.

Key insights

• Bayesian uncertainty quantification problems, specifically for linear models with Gaussian likelihoods and priors, can be reformulated and solved through the lens of viscous Hamilton-Jacobi Partial Differential Equations.
• The solutions to these viscous Hamilton-Jacobi PDEs can be precisely recovered via Riccati Ordinary Differential Equations, providing a computationally efficient and deterministic approach to obtain posterior mean and covariance.
• This PDE-based framework naturally supports dynamic learning paradigms, allowing for incremental model updates and continuous hyperparameter tuning without requiring full retraining or access to historical data.

Summary by alphaXiv.