A generative modeling / Physics-Informed Neural Network approach to random differential equations

2025-07-02

AI Summary

Modeling uncertainty in stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) remains challenging due to the interplay between physical constraints and probabilistic variability. Method: We propose a novel hybrid framework that deeply integrates generative models—specifically generative adversarial networks (GANs) and variational autoencoders (VAEs)—with physics-informed neural networks (PINNs). This is the first systematic incorporation of such generative priors into PINNs, enabling joint enforcement of physical laws and expressive probabilistic uncertainty representation. The framework jointly optimizes generative prior likelihood and physics-based residual loss to support efficient forward uncertainty propagation and controllable calibration. Results: Evaluated on diverse SDE/SPDE benchmarks, our method achieves significantly improved solution accuracy and yields well-calibrated, reliable uncertainty quantification. It extends the modeling capability and practical applicability of scientific machine learning for stochastic dynamical systems.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSocial Networks and Social Media: Generative AI / large language models and their impact on social systemsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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).

Problem

• Model uncertainty in complex systems using probabilistic frameworks
• Combine generative modeling with Physics-Informed Neural Networks
• Enhance uncertainty control in random differential equations

Innovation

• Physics-Informed Neural Networks with probabilistic frameworks
• Generative modeling combined with PINNs
• Uncertainty control in random differential equations