标题： 使用变分蒙特卡洛与神经网络估算 n-p 系统的逆散射势 显示英文标题

标题： Estimating Inverse Scattering Potentials for n-p System Using Variational Monte Carlo & Neural Networks

摘要： Riccati型非线性微分方程，也被称为变量相位法或相位函数法，用于构建氘核的 \( ^3S_1 \) 和 \( ^1S_0 \) 态的局域反势。 Morse 势通过调整参数使用变分蒙特卡洛（VMC）和多层感知器（MLP）型神经网络（NN）进行了优化。 从 VMC 和 NN 得到的反势表现出几乎相同的参数。 在 VMC 中，调整 Morse 势的所有三个参数以获得相移；而在 NN 中，三维参数优化问题被转化为一维参数优化问题，从而减少了优化参数的数量、时间和计算成本。 最近，GRANADA 组发布了一项散射数据的综合部分波分析，其中包括 1950 年至 2013 年间的 6713 个 \( np \) 相移数据点。 利用 GRANADA 的最终实验数据点，我们通过最小化均方误差（MSE）作为代价函数得到了 Morse 势的参数。发现对于 \( ^1S_0 \) 态，VMC（NN）的 MSE 为 0.65（2.5），对于 \( ^3S_1 \) 态，MSE 为 0.16（0.22）。 各种量子函数，如相位 \( \delta(r) \)、振幅 \( A(r) \)和波函数 \( u(r) \)，能量范围为 \( E_{\ell ab} = [1-350 \text{ MeV}] \)时，描述至 5 fm。 显示英文摘要

摘要： The Riccati-type nonlinear differential equation, also known as the Variable Phase Approach or Phase Function Method, is used to construct local inverse potentials for the \( ^3S_1 \) and \( ^1S_0 \) states of the deuteron. The Morse potential has been optimized by adjusting parameters using the Variational Monte Carlo (VMC) and Multilayer Perceptron (MLP) type Neural Networks (NN). The inverse potentials obtained from VMC and NN show almost identical parameters. In VMC, all three parameters of the Morse potential are varied to obtain the phase shifts, while in NN, the 3D-parameter optimization problem is converted to a 1D-parameter optimization problem, thus reducing optimization parameters, time, and computational cost. Recently, the GRANADA group published a comprehensive partial wave analysis of scattering data, which includes 6713 \( np \) phase shift data points from 1950 to 2013. Using the final experimental data points from GRANADA, we obtained the parameters for the Morse potential by minimizing the mean square error (MSE) as the cost function. The MSE using VMC (NN) is found to be 0.65 (2.5) for the \( ^1S_0 \) state and 0.16 (0.22) for the \( ^3S_1 \) state. Various quantum functions, such as phase \( \delta(r) \), amplitude \( A(r) \), and wave function \( u(r) \), are described up to 5 fm with energies \( E_{\ell ab} = [1-350 \text{ MeV}] \).