  Classical Physics-informed neural networks (PINNs) approximate solutions to
PDEs with the help of deep neural networks trained to satisfy the differential
operator and the relevant boundary conditions. We revisit this idea in the
quantum computing realm, using parameterised random quantum circuits as trial
solutions. We further adapt recent PINN-based techniques to our quantum
setting, in particular Gaussian smoothing. Our analysis concentrates on the
Poisson, the Heat and the Hamilton-Jacobi-Bellman equations, which are
ubiquitous in most areas of science. On the theoretical side, we develop a
complexity analysis of this approach, and show numerically that random quantum
networks can outperform more traditional quantum networks as well as random
classical networks.
