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.

在量子计算领域，我们使用参数化的随机量子电路作为试验解，将经典物理信息神经网络（PINN）的思想引入其中，以求解偏微分方程。我们进一步将近期基于 PINN 的技术适应于我们的量子设置，特别是高斯平滑。我们的分析集中在 Poisson 方程、热方程和 Hamilton-Jacobi-Bellman 方程上，这些方程在科学的大多数领域中都是普遍存在的。在理论方面，我们对这种方法进行了复杂性分析，并通过数值结果表明随机量子网络可超越传统量子网络和随机经典网络。