Neural Measures for learning distributions of Random PDEs

TL;DR

This paper introduces a probabilistic extension to Physics-Informed Neural Networks (PINNs) to better model uncertainty in solutions of random PDEs, combining generative models with PINNs for improved uncertainty quantification.

Contribution

It presents a novel integration of generative modeling with PINNs to systematically quantify uncertainty in solutions of random PDEs.

Findings

Enhanced uncertainty representation in forward problems.

Successful application to random differential equations.

Maintains predictive accuracy while controlling uncertainty.

Abstract

The integration of Scientific Machine Learning (SciML) techniques with uncertainty quantification (UQ) represents a rapidly evolving frontier in computational science. This work advances Physics-Informed Neural Networks (PINNs) by incorporating probabilistic frameworks to effectively model uncertainty in complex systems. Our approach enhances the representation of uncertainty in forward problems by combining generative modeling techniques with PINNs. This integration enables in a systematic fashion uncertainty control while maintaining the predictive accuracy of the model. We demonstrate the utility of this method through applications to random differential equations and random partial differential equations (PDEs).