  Unrolling training trajectories over time strongly influences the inference
accuracy of neural network-augmented physics simulators. We analyze these
effects by studying three variants of training neural networks on discrete
ground truth trajectories. In addition to commonly used one-step setups and
fully differentiable unrolling, we include a third, less widely used variant:
unrolling without temporal gradients. Comparing networks trained with these
three modalities makes it possible to disentangle the two dominant effects of
unrolling, training distribution shift and long-term gradients. We present a
detailed study across physical systems, network sizes, network architectures,
training setups, and test scenarios. It provides an empirical basis for our
main findings: A non-differentiable but unrolled training setup supported by a
numerical solver can yield 4.5-fold improvements over a fully differentiable
prediction setup that does not utilize this solver. We also quantify a
difference in the accuracy of models trained in a fully differentiable setup
compared to their non-differentiable counterparts. While differentiable setups
perform best, the accuracy of unrolling without temporal gradients comes
comparatively close. Furthermore, we empirically show that these behaviors are
invariant to changes in the underlying physical system, the network
architecture and size, and the numerical scheme. These results motivate
integrating non-differentiable numerical simulators into training setups even
if full differentiability is unavailable. We also observe that the convergence
rate of common neural architectures is low compared to numerical algorithms.
This encourages the use of hybrid approaches combining neural and numerical
algorithms to utilize the benefits of both.
