Enforcing Imprecise Constraints on Generative Adversarial Networks for Emulating Physical Systems

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.

From data to insight, enhancing structural health monitoring using physics-informed machine learning and advanced data collection methods

“…As a result, introducing physical limitations to the generator loss function gives an alternate method for dealing with the vanishing gradient problem in GAN training. Moreover, adding a physics penalty gives an extra edge to the generator which accelerates its convergence and helps the discriminator achieve equilibrium [149]. Figure 10 represents the block diagram for PI-GAN.…”

Multiscale Neural Operator: Learning Fast and Grid-independent PDE Solvers

“…Our approach leverages known equations for computing a coarse-grid prior; which is complementary to using known equations as soft [109,74,142,137,144,139] or hard constraints [50,89,8,39,7,61] as these methods can still be used to constrain the learned parametrization. In terms of symmetry, our approach exploits translational equivariance via Fourier transformations [76], but can be extended to other frameworks that exploit in-or equivariance of PDEs [95] to rotational [44,124], Galilean [136,105], scale [9], translational [123], reflectional [37] or permutational [145] transformations.…”