Enforcing Deterministic Constraints on Generative Adversarial Networks for Emulating Physical Systems

Generative adversarial networks (GANs) are 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 in turbulent flows). The latter have been studied in a companion paper (Wu et al. 2019. In the present work, we enforce deterministic yet approximate 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 precisely conform to the underlying constraints, even though the constraints are only enforced approximately. More importantly, the imposed constraints significantly accelerate the convergence and improve the robustness in the training. These improvements are noteworthy, as the convergence and robustness are two well-known obstacles in the training of GANs. Keywords: Generative adversarial networks, physics constraints, physics-informed machine learning 1. Introduction Machine learning and particularly deep learning has achieved significant success in a wide range of commercial domain applications such as image recognition, audio recognition, and natural language processing [1-5]. Corresponding author Email address: hengxiao@vt.edu For example, machine learning methods such as random forests and neural networks have been used to provide closure models for turbulent flows [6-9] and multiphase flows [10, 11] and to compute rock permeability directly from CT scan images [12]. They have also been used to discover ordinary and partial differential equations (ODEs and PDEs) from data [13-16]. Finally, neural networks have been used to solve exactly specified PDEs [17-20] and partially known PDEs by incorporating available data [21-24]. The scientific applications reviewed above mostly involve supervised learning, which consists of three steps: (a) postulate a model that maps inputs (features) to outputs (labels), controlled by a set of adjustable model parameters; (b) learn the parameters from training data (labeled examples of input-output pairs); and (c) use the fitted model to predict the responses for new inputs that were not included in the training data.