| Abstract: | Generative adversarial networks (GANs) were initially proposed to generate images by learning from a large number of samples. Recently, GANs have been
used to emulate complex physical systems such as turbulent flows. However, a critical question must be answered before GANs can be considered trusted emulators for
physical systems: do GANs-generated samples conform to the various physical constraints? These include both deterministic constraints (e.g., conservation laws) and
statistical constraints (e.g., energy spectrum of turbulent flows). The latter have been
studied in a companion paper (Wu et al., Enforcing statistical constraints in generative
adversarial networks for modeling chaotic dynamical systems. Journal of Computational Physics. 406, 109209, 2020). In the present work, we enforce deterministic yet
imprecise constraints on GANs by incorporating them into the loss function of the
generator. 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. In
both cases, the constrained GANs produced samples that conform to the underlying
constraints rather accurately, even though the constraints are only enforced up to a
specified interval. More importantly, the imposed constraints significantly accelerate
the convergence and improve the robustness in the training, indicating that they serve
as a physics-based regularization. These improvements are noteworthy, as the convergence and robustness are two well-known obstacles in the training of GANs. |
