Enforcing Physical Constraints in CNNs through Differentiable PDE Layer

Recent studies at the intersection of physics and deep learning have illustrated successes in the application of deep neural networks to partially or fully replace costly physics simulations. Enforcing physical constraints to solutions generated by neural networks remains a challenge, yet it is essential to the accuracy and trustworthiness of such model predictions. Many systems in the physical sciences are governed by Partial Differential Equations (PDEs). Enforcing these as hard constraints, we show, are inefficient in conventional frameworks due to the high dimensionality of the generated fields. To this end, we propose the use of a novel differentiable spectral projection layer for neural networks that efficiently enforces spatial PDE constraints using spectral methods, yet is fully differentiable, allowing for its use as a layer within Convolutional Neural Networks (CNNs) during end-to-end training. We show that its computational cost is cheaper than a single convolution layer. We apply it to an important class of physical systems - incompressible turbulent flows, where the divergence-free PDE constraint is required. We train a 3D Conditional Generative Adversarial Network (CGAN) for turbulent flow superresolution efficiently, while guaranteeing the spatial PDE constraint of zero divergence. Furthermore, our empirical results show that the model produces realistic flow statistics when trained with hard constraints imposed via the proposed novel differentiable spectral projection layer, as compared to soft constrained and unconstrained counterparts.