| Abstract: | Generative adversarial networks (GANs) were initially proposed to generate images by learning from a large number of samples. Recently, GANs have beenused to emulate complex physical systems such as turbulent flows. However, a critical question must be answered before GANs can be considered trusted emulators forphysical systems: do GANs-generated samples conform to the various physical constraints? These include both deterministic constraints (e.g., conservation laws) andstatistical constraints (e.g., energy spectrum of turbulent flows). The latter have beenstudied in a companion paper (Wu et al., Enforcing statistical constraints in generativeadversarial networks for modeling chaotic dynamical systems. Journal of Computational Physics. 406, 109209, 2020). In the present work, we enforce deterministic yetimprecise constraints on GANs by incorporating them into the loss function of thegenerator. We evaluate the performance of physics-constrained GANs on two representative tasks with geometrical constraints (generating points on circles) and differential constraints (generating divergence-free flow velocity fields), respectively. Inboth cases, the constrained GANs produced samples that conform to the underlyingconstraints rather accurately, even though the constraints are only enforced up to aspecified interval. More importantly, the imposed constraints significantly acceleratethe convergence and improve the robustness in the training, indicating that they serveas a physics-based regularization. These improvements are noteworthy, as the convergence and robustness are two well-known obstacles in the training of GANs. |
