Competitive Physics Informed Networks

Physics Informed Neural Networks (PINNs) solve partial differential equations (PDEs) by representing them as neural networks. The original PINN implementation does not provide high accuracy, typically attaining about $10^{-3}$ $L_2$ relative error. We formulate and test an adversarial approach called competitive PINNs (CPINNs) to overcome this limitation. CPINNs train a discriminator that is rewarded for predicting PINN mistakes. The discriminator and PINN participate in a zero-sum game with the exact PDE solution as an optimal strategy. This approach avoids the issue of squaring the large condition numbers of PDE discretizations. Numerical experiments show that a CPINN trained with competitive gradient descent can achieve errors two orders of magnitude smaller than that of a PINN trained with Adam or stochastic gradient descent.