  Neural networks can be trained to solve partial differential equations (PDEs)
by using the PDE residual as the loss function. This strategy is called
"physics-informed neural networks" (PINNs), but it currently cannot produce
high-accuracy solutions, typically attaining about $0.1\%$ relative error. We
present an adversarial approach that overcomes this limitation, which we call
competitive PINNs (CPINNs). CPINNs train a discriminator that is rewarded for
predicting mistakes the PINN makes. The discriminator and PINN participate in a
zero-sum game with the exact PDE solution as an optimal strategy. This approach
avoids squaring the large condition numbers of PDE discretizations, which is
the likely reason for failures of previous attempts to decrease PINN errors
even on benign problems. Numerical experiments on a Poisson problem show that
CPINNs achieve errors four orders of magnitude smaller than the best-performing
PINN. We observe relative errors on the order of single-precision accuracy,
consistently decreasing with each epoch. To the authors' knowledge, this is the
first time this level of accuracy and convergence behavior has been achieved.
Additional experiments on the nonlinear Schr\"odinger, Burgers', and Allen-Cahn
equation show that the benefits of CPINNs are not limited to linear problems.

Machine Learning;Multiagent Systems;Numerical Analysis;Numerical Analysis;Optimization and Control