  We present a convolutional framework which significantly reduces the
complexity and thus, the computational effort for distributed reinforcement
learning control of dynamical systems governed by partial differential
equations (PDEs). Exploiting translational invariances, the high-dimensional
distributed control problem can be transformed into a multi-agent control
problem with many identical, uncoupled agents. Furthermore, using the fact that
information is transported with finite velocity in many cases, the dimension of
the agents' environment can be drastically reduced using a convolution
operation over the state space of the PDE. In this setting, the complexity can
be flexibly adjusted via the kernel width or by using a stride greater than
one. Moreover, scaling from smaller to larger systems -- or the transfer
between different domains -- becomes a straightforward task requiring little
effort. We demonstrate the performance of the proposed framework using several
PDE examples with increasing complexity, where stabilization is achieved by
training a low-dimensional deep deterministic policy gradient agent using
minimal computing resources.
