In this work, we solve differential equations using quantum Chebyshev feature
maps. We propose a tensor product over a summation of Pauli-Z operators as a
change in the measurement observables resulting in improved accuracy and
reduced computation time for initial value problems processed by floating
boundary handling. This idea has been tested on solving the complex dynamics of
a Riccati equation as well as on a system of differential equations.
Furthermore, a second-order differential equation is investigated in which we
propose adding entangling layers to improve accuracy without increasing the
variational parameters. Additionally, a modified self-adaptivity approach of
physics-informed neural networks is incorporated to balance the multi-objective
loss function. Finally, a new quantum circuit structure is proposed to
approximate multivariable functions, tested on solving a 2D Poisson's equation.

用量子 Chebyshev 特征映射来解微分方程，通过求和保里 Z 算符的张量积作为测量可观测量的改变，提高了精度和计算时间，在处理初始值问题时使用浮动边界处理。在复杂动力学和微分方程系统的求解上进行了测试，另外，我们提出了加入纠缠层以提高精度而不增加可变参数的方案。此外，还结合了物理信息神经网络的改进自适应方法来平衡多目标损失函数。最后，提出了一种新的量子电路结构来逼近多变量函数，并测试了在求解 2D Poisson 方程时的效果。