Solving partial differential equations (PDEs) has been a fundamental problem
in computational science and of wide applications for both scientific and
engineering research. Due to its universal approximation property, neural
network is widely used to approximate the solutions of PDEs. However, existing
works are incapable of solving high-order PDEs due to insufficient calculation
accuracy of higher-order derivatives, and the final network is a black box
without explicit explanation. To address these issues, we propose a deep
learning framework to solve high-order PDEs, named SHoP. Specifically, we
derive the high-order derivative rule for neural network, to get the
derivatives quickly and accurately; moreover, we expand the network into a
Taylor series, providing an explicit solution for the PDEs. We conduct
experimental validations four high-order PDEs with different dimensions,
showing that we can solve high-order PDEs efficiently and accurately.

本文提出了一种名为 SHoP 的深度学习框架，通过推导神经网络的高阶导数规则和将网络扩展为泰勒级数，实现了高阶偏微分方程的快速准确求解，并通过实验验证了其有效性。

SHoP：用于求解高阶偏微分方程的深度学习框架

SHoP: A Deep Learning Framework for Solving High-order Partial  Differential Equations

Quantum neural networks (QNNs) offer a powerful paradigm for programming
near-term quantum computers and have the potential to speedup applications
ranging from data science to chemistry to materials science. However, a
possible obstacle to realizing that speedup is the Barren Plateau (BP)
phenomenon, whereby the gradient vanishes exponentially in the system size $n$
for certain QNN architectures. The question of whether high-order derivative
information such as the Hessian could help escape a BP was recently posed in
the literature. Here we show that the elements of the Hessian are exponentially
suppressed in a BP, so estimating the Hessian in this situation would require a
precision that scales exponentially with $n$. Hence, Hessian-based approaches
do not circumvent the exponential scaling associated with BPs. We also show the
exponential suppression of higher order derivatives. Hence, BPs will impact
optimization strategies that go beyond (first-order) gradient descent. In
deriving our results, we prove novel, general formulas that can be used to
analytically evaluate any high-order partial derivative on quantum hardware.
These formulas will likely have independent interest and use for training
quantum neural networks (outside of the context of BPs).

本研究探讨了量子神经网络中的 Barren Plateau 现象，研究发现高阶导数信息无法解决 Barren Plateau 的指数级灭减问题，因此超越（一阶）梯度下降的优化策略仍会受到 Barren Plateau 的影响。同时，我们证明了新颖的通用公式，可用于在量子硬件上分析求解任何高阶偏导数，这些公式可能有独立的兴趣和用途，可用于量子神经网络的训练（除了 Barren Plateau 的情境）。