Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the potential for enhancing a classical deep learning-based method for solving high-dimensional nonlinear partial differential equations with suitable quantum subroutines. First, with near-term noisy intermediate-scale quantum computers in mind, we construct architectures employing variational quantum circuits and classical neural networks in conjunction. While the hybrid architectures show equal or worse performance than their fully classical counterparts in simulations, they may still be of use in very high-dimensional cases or if the problem is of a quantum mechanical nature. Next, we identify the bottlenecks imposed by Monte Carlo sampling and the training of the neural networks. We find that quantum-accelerated Monte Carlo methods, as well as classical multi-level Monte Carlo methods, offer the potential to speed up the estimation of the loss function. In addition, we identify and analyse the trade-offs when using quantum-accelerated Monte Carlo methods to estimate the gradients with different methods, including a recently-developed back propagation-free forward gradient method. Finally, we discuss the usage of a suitable quantum algorithm for accelerating the training of feed-forward neural networks. Hence, this work provides different avenues with the potential for polynomial speedups for deep learning-based methods for nonlinear partial differential equations. 中文翻译： 偏微分方程经常出现在自然科学和相关学科中。由于“维度灾难”，解决它们通常具有挑战性，尤其是在高维度中。在这项工作中，我们探索了增强基于经典深度学习的方法的潜力，该方法用于使用合适的量子子程序求解高维非线性偏微分方程。首先，考虑到近期嘈杂的中型量子计算机，我们构建了结合使用变分量子电路和经典神经网络的架构。虽然混合架构在模拟中显示出与完全经典的对应架构相同或更差的性能，但它们可能仍可用于非常高维的情况，或者如果问题具有量子力学性质。下一个，我们确定了蒙特卡洛采样和神经网络训练带来的瓶颈。我们发现量子加速蒙特卡罗方法以及经典的多级蒙特卡罗方法提供了加速损失函数估计的潜力。此外，我们确定并分析了使用量子加速蒙特卡洛方法估计梯度时的权衡，包括最近开发的无反向传播前向梯度法。最后，我们讨论了使用合适的量子算法来加速前馈神经网络的训练。因此，这项工作为基于深度学习的非线性偏微分方程方法的多项式加速提供了不同的途径。