In this paper, we study a physics-informed algorithm for Wasserstein Generative Adversarial Networks (WGANs) for uncertainty quantification in solutions of partial differential equations. By using groupsort activation functions in adversarial network discriminators, network generators are utilized to learn the uncertainty in solutions of partial differential equations observed from the initial/boundary data. Under mild assumptions, we show that the generalization error of the computed generator converges to the approximation error of the network with high probability, when the number of samples are sufficiently taken. According to our established error bound, we also find that our physics-informed WGANs have higher requirement for the capacity of discriminators than that of generators. Numerical results on synthetic examples of partial differential equations are reported to validate our theoretical results and demonstrate how uncertainty quantification can be obtained for solutions of partial differential equations and the distributions of initial/boundary data.
翻译:在本文中,我们研究了瓦塞尔斯坦基因反转网络(WGANs)的物理知情算法,用于在局部差异方程式的解决方案中进行不确定性量化。通过在对抗性网络歧视者中使用群集激活功能,网络生成器被用来学习从初始/边界数据中观察到的局部差异方程式解决方案的不确定性。在温和假设下,我们显示计算生成器的概括误差与网络近似误差非常接近,当样本数量充足时。根据我们确定的误差,我们还发现我们的物理学知情的WGAN对歧视者能力的要求高于对生成者的要求。据报告,部分差异方程式合成实例的数值结果证实了我们的理论结果,并表明如何为部分差异方程式的解决方案和初始/边界数据的分布而获得不确定性的量化。