  The use of deep learning methods in scientific computing represents a
potential paradigm shift in engineering problem solving. One of the most
prominent developments is Physics-Informed Neural Networks (PINNs), in which
neural networks are trained to satisfy partial differential equations (PDEs).
While this method shows promise, the standard version has been shown to
struggle in accurately predicting the dynamic behavior of time-dependent
problems. To address this challenge, methods have been proposed that decompose
the time domain into multiple segments, employing a distinct neural network in
each segment and directly incorporating continuity between them in the loss
function of the minimization problem. In this work we introduce a method to
exactly enforce continuity between successive time segments via a solution
ansatz. This hard constrained sequential PINN (HCS-PINN) method is simple to
implement and eliminates the need for any loss terms associated with temporal
continuity. The method is tested for a number of benchmark problems involving
both linear and non-linear PDEs. Examples include various first order time
dependent problems in which traditional PINNs struggle, namely advection,
Allen-Cahn, and Korteweg-de Vries equations. Furthermore, second and third
order time-dependent problems are demonstrated via wave and Jerky dynamics
examples, respectively. Notably, the Jerky dynamics problem is chaotic, making
the problem especially sensitive to temporal accuracy. The numerical
experiments conducted with the proposed method demonstrated superior
convergence and accuracy over both traditional PINNs and the soft-constrained
counterparts.
