Summary of Leveraging Viscous Hamilton-jacobi Pdes For Uncertainty Quantification in Scientific Machine Learning, by Zongren Zou et al.
Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning
by Zongren Zou, Tingwei Meng, Paula Chen, Jérôme Darbon, George Em Karniadakis
First submitted to arxiv on: 12 Apr 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Machine Learning (stat.ML)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The paper proposes a new framework for uncertainty quantification (UQ) in scientific machine learning (SciML), combining predictive power with reliability measures. It addresses two major challenges: limited interpretability and expensive training procedures. The authors establish a connection between Bayesian inference problems in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). They show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to these HJ PDEs. The paper specializes in Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors, solving associated viscous HJ PDEs using Riccati ODEs. A Riccati-based methodology is developed, providing computational advantages for continuous model updates. This approach can efficiently add or remove data points and tune hyperparameters without retraining on previous data. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper is about how to make machine learning models more reliable by quantifying the uncertainty in their predictions. It finds a new way to connect two different areas of research: scientific machine learning (SciML) and mathematical equations called Hamilton-Jacobi partial differential equations (HJ PDEs). This connection allows them to solve problems in SciML more efficiently, which is important for making decisions based on uncertain data. |
Keywords
* Artificial intelligence * Bayesian inference * Machine learning
