SchrödingerNet：一种用于求解薛定谔方程的通用神经网络求解器
SchrödingerNet: A Universal Neural Network Solver for The Schrödinger Equation
AI总结：
针对现有基于神经网络的薛定谔方程求解依赖玻恩-奥本海默近似、单构型训练成本高的问题，提出SchrödingerNet架构，通过对称适配波函数拟设和局部能量均衡损失函数，单次训练即可求解全电子-核薛定谔方程并生成势能面，经基准测试验证了准确性与高效性。
摘要：
机器学习的最新进展通过将多种基于神经网络（NN）的波函数拟设与变分蒙特卡洛方法相结合，推动了电子薛定谔方程（SE）的高精度数值求解。然而，这类基于神经网络的方法全部依赖玻恩-奥本海默近似（BOA），且针对每一种核构型都需要进行计算成本高昂的训练。本研究中，我们提出了一种新型神经网络架构——SchrödingerNet，通过设计旨在均衡系统各处局部能量的损失函数，求解完整的电子-核薛定谔方程。该方法基于具有平移、旋转和置换对称性适配的总波函数拟设，同时包含核坐标与电子坐标。这一策略不仅能在采样充分的核构型空间内，针对任意几何构型高效、准确地生成连续势能面，还能通过单次训练过程纳入非玻恩-奥本海默近似修正。与原子和小分子系统基准的对比结果验证了该方法的准确性与高效性。
英文摘要：
Recent advances in machine learning have facilitated numerically accurate solution of the electronic Schrödinger equation (SE) by integrating various neural network (NN)-based wavefunction ansatzes with variational Monte Carlo methods. Nevertheless, such NN-based methods are all based on the Born-Oppenheimer approximation (BOA) and require computationally expensive training for each nuclear configuration. In this work, we propose a novel NN architecture, SchrödingerNet, to solve the full electronic-nuclear SE by defining a loss function designed to equalize local energies across the system. This approach is based on a translationally, rotationally and permutationally symmetry-adapted total wavefunction ansatz that includes both nuclear and electronic coordinates. This strategy not only allows for an efficient and accurate generation of a continuous potential energy surface at any geometry within the well-sampled nuclear configuration space, but also incorporates non-BOA corrections, through a single training process. Comparison with benchmarks of atomic and small molecular systems demonstrates its accuracy and efficiency.
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