  Standard approaches to controlling dynamical systems involve biologically
implausible steps such as backpropagation of errors or intermediate model-based
system representations. Recent advances in machine learning have shown that
"imperfect" feedback of errors during training can yield test performance that
is similar to using full backpropagated errors, provided that the two error
signals are at least somewhat aligned. Inspired by such methods, we introduce
an iterative, spatiotemporally local protocol to learn driving forces and
control non-equilibrium dynamical systems using imperfect feedback signals. We
present numerical experiments and theoretical justification for several
examples. For systems in conservative force fields that are driven by external
time-dependent protocols, our update rules resemble a dynamical version of
contrastive divergence. We appeal to linear response theory to establish that
our imperfect update rules are locally convergent for these conservative
systems. For systems evolving under non-conservative dynamics, we derive a new
theoretical result that makes possible the control of non-equilibrium
steady-state probabilities through simple local update rules. Finally, we show
that similar local update rules can also solve dynamical control problems for
non-conservative systems, and we illustrate this in the non-trivial example of
active nematics. Our updates allow learning spatiotemporal activity fields that
pull topological defects along desired trajectories in the active nematic
fluid. These imperfect feedback methods are information efficient and in
principle biologically plausible, and they can help extend recent methods of
decentralized training for physical materials into dynamical settings.
