  Motivated by the recent developments in artificial intelligence, we introduce
linear quadratic deep structured teams in this paper. Two notions of
equivariant and partially equivariant systems are defined, and it is shown that
such systems can be partitioned into a few sub-populations of decision makers,
where every decision maker in each sub-population is coupled in both dynamics
and cost function through a set of linear regressions of the states and actions
of all decision makers. Two non-classical information structures are
considered: deep-state sharing and partial deep-state sharing, where deep state
refers to the linear regression of the states of the decision makers in each
sub-population. For a risk-sensitive cost function with deep-state sharing
structure, a closed-form low-complexity representation of the globally optimal
strategy is obtained, whose computational complexity is independent of the
number of decision makers in each sub-population. In addition, it is shown that
the risk-sensitive solution converges to the risk-neutral one as the number of
decision makers increases to infinity. Moreover, two sub-optimal sequential
strategies under partial deep-state sharing information structure are proposed
by introducing two Kalman-like filters, one based on the finite-population
model and the other one based on the infinite-population model. It is proved
that the prices of information associated with the above sub-optimal solutions
converge to zero as the number of decision makers goes to infinity.
Furthermore, a class of feed-forward deep neural networks with multiple layers
of weighted sums and products is introduced wherein the optimal weights and
biases are explicitly obtained. A supply-chain management example is presented
to demonstrate the efficacy of the obtained results.
