Gen-PINNs: Generative Adversarial Physics Informed Neural Networks for solving partial differential equations

Physics Informed Neural Networks (PINNs), have been the standard data-free method for solving Partial Differential Equations (PDEs), using machine learning. With recent advances in Generative Adversarial Networks (GANs), many apply them for solving complex data-driven problems. However, the use of PINNs in GAN-based approaches is limited. In this work, we first identify limitations in the standard PINNs method for solving PDEs. Next, we combine and test multiple PINN extensions to overcome these limitations in a GAN architecture and then construct a novel Generative Adversarial Physics Informed Neural Network (Gen-PINN) framework to solve sharp- or shock-front bearing PDEs. In this approach, the generator learns the underlying PDE solution using dynamically weighted different loss components while the discriminator discriminates between the real known solution, at initial, boundary, and collocation points, against the generator output. Furthermore, several PINNs extensions are explored and combined inside the model to leverage the unique solution-finding ability of each method. Aspects are investigated, including: The objective loss functions of the generator and discriminator are specified by the type of loss and the loss schedule for the generator and discriminator, a variational form of mean square loss function, optimization algorithms, dynamic loss weighting, and Fourier input embeddings. The various methods are incorporated in the Gen-PINNs method and tested against standard PINNs to demonstrate effectiveness in using a synergistic unified approach to solve non-linear and higher-order PDEs. The improvements are demonstrated by testing complex PDEs, including Burgers’, advection-diffusion, Allen-Cahn, and the Kuramoto-Sivashinsky PDEs. The approach highlights the potential of improving shock-front PDE modeling by combining PINN extensions using Gen-PINNs.