  Deep learning-based surrogate modeling is becoming a promising approach for
learning and simulating dynamical systems. Deep-learning methods, however, find
very challenging learning stiff dynamics. In this paper, we develop DAE-PINN,
the first effective deep-learning framework for learning and simulating the
solution trajectories of nonlinear differential-algebraic equations (DAE),
which present a form of infinite stiffness and describe, for example, the
dynamics of power networks. Our DAE-PINN bases its effectiveness on the synergy
between implicit Runge-Kutta time-stepping schemes (designed specifically for
solving DAEs) and physics-informed neural networks (PINN) (deep neural networks
that we train to satisfy the dynamics of the underlying problem). Furthermore,
our framework (i) enforces the neural network to satisfy the DAEs as
(approximate) hard constraints using a penalty-based method and (ii) enables
simulating DAEs for long-time horizons. We showcase the effectiveness and
accuracy of DAE-PINN by learning and simulating the solution trajectories of a
three-bus power network.
