Sequential-in-time training of nonlinear parametrizations for solving
  time-dependent partial differential equations
Huan Zhang Yifan Chen Eric Vanden-Eijnden Benjamin Peherstorfer
Sequential-in-time methods solve a sequence of training problems to fit
nonlinear parametrizations such as neural networks to approximate solution
trajectories of partial differential equations over time. This work shows that
sequential-in-time training methods can be understood broadly as either
optimize-then-discretize (OtD) or discretize-then-optimize (DtO) schemes, which
are well known concepts in numerical analysis. The unifying perspective leads
to novel stability and a posteriori error analysis results that provide
insights into theoretical and numerical aspects that are inherent to either OtD
or DtO schemes such as the tangent space collapse phenomenon, which is a form
of over-fitting. Additionally, the unified perspective facilitates establishing
connections between variants of sequential-in-time training methods, which is
demonstrated by identifying natural gradient descent methods on energy
functionals as OtD schemes applied to the corresponding gradient flows.