发表机构
University of the Punjab; Universidad Michoacana de San Nicolás de Hidalgo; Universidad del Bío-Bío（旁遮普大学; 米却肯州立圣尼古拉斯德伊达尔戈大学; 比奥比奥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将物理信息神经网络与量子神经网络应用于一维薛定谔本征值问题，在谐振子和方势阱基准上高精度复现本征值，且量子电路在高激发态收敛更可靠。
AI 中文摘要
一维空间中的薛定谔方程允许存在少量精确可解的势，这些势为任何新的本征值求解器提供了天然的验证平台。我们针对不含时薛定谔方程构建了物理信息神经网络（PINNs）和物理信息量子神经网络（PIQNNs），并将其应用于三个基准问题：谐振子、无限深方势阱和有限深方势阱。在每种情况下，复合损失函数编码了微分方程残差、归一化条件、边界行为以及本征态之间的正交性，从而在没有监督数据的情况下，使试探波函数被驱动向真正的本征函数。两种方法返回的本征值和波函数与精确谱以及三种经典参考方法（矩阵Numerov法、有限差分法和打靶法）进行了比较。对于光滑的谐振子，两种神经求解器将最低四个本征值复现到百万分之一精度，而对于方势阱，即使在势不连续的情况下，它们也能以相当的保真度恢复解析能级。量子电路采用分层角度嵌入拟设并带有强纠缠块，在高激发态上比经典对应方法更可靠地收敛，而在这些高激发态上，经典网络的损失景观变得难以导航。
英文摘要
The Schrodinger equation in one spatial dimension admits a small set of exactly solvable potentials that serve as natural proving grounds for any new eigenvalue solver. We formulate Physics-Informed Neural Networks (PINNs) and Physics-Informed Quantum Neural Networks (PIQNNs) for the time-independent Schrodinger equation and apply them to three of these benchmarks: the harmonic oscillator, the infinite square well, and the finite square well. In each case a composite loss encodes the differential-equation residual, the normalization condition, the boundary behavior, and the orthogonality between eigenstates, so that the trial wave function is driven toward a genuine eigenfunction without supervised data. The eigenvalues and wave functions returned by both methods are compared against the exact spectra and against three classical references: the matrix Numerov method, the finite difference method, and the shooting method. For the smooth oscillator the two neural solvers reproduce the lowest four eigenvalues to parts per million, while for the square wells they recover the analytic levels with comparable fidelity even where the potential is discontinuous. The quantum circuit, built as a layered angle-embedding ansatz with strongly entangling blocks, converges more reliably than its classical counterpart on the higher excited states, where the loss landscape of the classical network becomes harder to navigate.
