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).

翻译：科学机器学习（SciML）技术与不确定性量化（UQ）的融合代表了计算科学中一个快速发展的前沿领域。本研究通过引入概率框架来改进物理信息神经网络（PINNs），从而有效建模复杂系统中的不确定性。我们的方法将生成建模技术与PINNs相结合，增强了前向问题中不确定性的表征能力。这种集成能够在保持模型预测精度的同时，以系统化的方式实现不确定性控制。我们通过随机微分方程和随机偏微分方程（PDEs）的应用实例，验证了该方法的实用性。