Leveraging Viscous Hamilton-Jacobi PDEs for Efficient Uncertainty Quantification in Scientific Machine Learning

Core Concepts

Viscous Hamilton-Jacobi partial differential equations can be leveraged to efficiently solve Bayesian inference problems arising in scientific machine learning, enabling continuous model updates, hyperparameter tuning, and uncertainty quantification.

Abstract

The content presents a new theoretical connection between certain Bayesian inference problems in scientific machine learning (SciML) and viscous Hamilton-Jacobi (HJ) partial differential equations (PDEs). Specifically, the authors show that the posterior mean and covariance in Bayesian inference can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE.
As a first exploration of this connection, the authors specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, leading to a new Riccati-based methodology. This Riccati-based approach provides several potential computational advantages:

It can efficiently add or remove data points to the training set without retraining on or accessing the previously incorporated data, and this update is invariant to the order of the data.
It provides a continuous flow of solutions with respect to the hyperparameters of the model, allowing the hyperparameters to be tuned continuously.
It can handle different learning scenarios, such as big data and active learning settings, by leveraging the uncertainty metrics to inform the learning process.

The authors demonstrate the potential benefits of this Riccati-based approach through several examples from SciML involving noisy data and epistemic uncertainty. The results show that the predicted uncertainty metrics are good indicators of the reliability of the learned models.

Stats

The exact solution to the boundary value problem (4.2) is u(τ) = exp(-2τ) sin(15τ).
The data of u0, uT are corrupted by additive Gaussian noise with mean zero and standard deviation 0.01.
The data of u'0, u'T are corrupted by additive Gaussian noise with mean zero and standard deviation 0.001.
The data of the source term f is corrupted by additive Gaussian noise.

Quotes

"Viscous Hamilton-Jacobi partial differential equations can be leveraged to efficiently solve Bayesian inference problems arising in scientific machine learning, enabling continuous model updates, hyperparameter tuning, and uncertainty quantification."
"The Riccati-based approach can efficiently add or remove data points to the training set without retraining on or accessing the previously incorporated data, and this update is invariant to the order of the data."
"The Riccati-based approach provides a continuous flow of solutions with respect to the hyperparameters of the model, allowing the hyperparameters to be tuned continuously."

Key Insights Distilled From

Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning

by Zong... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.08809.pdf

Deeper Inquiries

How can the Riccati-based methodology be extended to handle more general Bayesian inference problems beyond linear models and Gaussian likelihoods/priors

The Riccati-based methodology can be extended to handle more general Bayesian inference problems by considering non-linear models and non-Gaussian likelihoods/priors.
For non-linear models, the linear relationship between the model parameters and the data can be generalized to non-linear functions. This would involve updating the Riccati ODEs to accommodate the non-linear transformations of the model parameters. The key would be to find a suitable representation of the non-linear model that allows for efficient computation of the posterior mean and covariance.
Similarly, for non-Gaussian likelihoods and priors, the methodology can be adapted by incorporating the appropriate likelihood and prior distributions into the framework. This may involve modifying the initial conditions of the Riccati ODEs to reflect the characteristics of the non-Gaussian distributions and updating the equations accordingly.
In essence, extending the Riccati-based methodology to handle more general Bayesian inference problems involves adapting the framework to accommodate the specific characteristics of the non-linear models and non-Gaussian distributions involved in the problem.

What are the potential limitations or drawbacks of the Riccati-based approach, and how can they be addressed

One potential limitation of the Riccati-based approach is its reliance on the assumption of linearity in the model and Gaussian distributions for the likelihood and prior. This restricts its applicability to a specific subset of Bayesian inference problems and may not capture the complexity of real-world data that often exhibit non-linear relationships and non-Gaussian distributions.
To address this limitation, one approach could be to explore the use of non-linear transformations or kernel methods to model more complex relationships in the data. By incorporating non-linearities into the model, the Riccati-based methodology can be extended to handle a wider range of Bayesian inference problems.
Another limitation could be the computational complexity of solving the Riccati ODEs, especially for high-dimensional problems. This can be addressed by exploring more efficient numerical methods or approximations that can reduce the computational burden while maintaining accuracy in the inference results.
Additionally, the Riccati-based approach may struggle with handling outliers or noisy data, as it assumes a certain level of noise in the data. Techniques such as robust estimation or outlier detection methods can be integrated into the methodology to improve its robustness in the presence of noisy data.

Can the connection between viscous Hamilton-Jacobi PDEs and Bayesian inference be leveraged to develop new uncertainty quantification techniques for other machine learning models beyond scientific machine learning

The connection between viscous Hamilton-Jacobi PDEs and Bayesian inference can indeed be leveraged to develop new uncertainty quantification techniques for other machine learning models beyond scientific machine learning.
For example, in the field of natural language processing, where models often deal with complex linguistic structures and uncertainties, the principles of uncertainty quantification derived from the connection can help in assessing the reliability of language models and improving their interpretability.
In computer vision, where image data can be noisy and ambiguous, leveraging the uncertainty quantification techniques inspired by the connection can aid in understanding the confidence levels of object detection or image classification models.
Furthermore, in reinforcement learning, where agents interact with environments and make decisions based on uncertain information, incorporating uncertainty quantification methods based on the connection can enhance the decision-making process and improve the stability of learning algorithms.
By applying the principles of uncertainty quantification derived from the connection to a broader range of machine learning models, researchers can develop more robust and reliable systems that are capable of handling uncertainties inherent in real-world data and applications.