Solution of the many-electron Schrödinger equation with deep neural networks

如果能得到多电子薛定谔方程的精确解，几乎所有的化学都可以从第一原理中得到。有趣的化学体系的精确波函数是遥不可及的，因为它们通常是NP-难计算的，但可以使用多项式标度算法找到近似。对于这些算法中的许多算法来说，关键挑战是选择近似波函数，这必须在效率和精度之间找到平衡。神经网络作为精确实用的函数逼近器已经显示出令人印象深刻的能力，并有望成为自旋系统的紧凑近似波函数，但电子结构问题要求波函数服从费米-狄拉克统计。在最近的预印本(arxiv：1909.02487)中，我们介绍了一种新的深度学习结构，费米子神经网络，作为一个强大的多电子系统的近似波函数，并表明这种方法可以与众所周知的简单的变分量子蒙特卡罗方法相结合，获得远远超过以前在原子和小分子变分量子蒙特卡罗模拟中所达到的精度。使用原子位置和电荷以外的数据，我们预测了氮分子和氢链这两个具有挑战性的强关联系统的解离曲线，预测精度明显高于耦合团簇方法，后者被广泛认为是量子化学在平衡几何条件下最精确的可伸缩方法。这表明，深度神经网络可以提高变分量子蒙特卡罗方法的精度，甚至超过其他从头算量子化学方法。卡塞拉的博士项目将以这一有希望的开始为基础，使用神经网络试验波函数来研究简单固体。为了简单起见，我们将从研究均匀的电子气开始(这将需要大量的代码开发)，然后再转到固体氢，也许还有其他简单的固体。我们还将研究使用费米子神经网络来提高更复杂的扩散量子蒙特卡罗方法的精度。

Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles. Exact wavefunctions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially-scaling algorithms. The key challenge for many of these algorithms is the choice of approximate wavefunction, which must find a balance between efficiency and accuracy. Neural networks have shown impressive power as accurate practical function approximators and promise as a compact approximate wavefunction for spin systems, but problems in electronic structure require wavefunctions that obey Fermi-Dirac statistics. In a very recent preprint (arXiv:1909.02487), we introduced a novel deep learning architecture, the Fermionic Neural Network, as a powerful approximate wavefunction for many-electron systems and showed that this can be combined with the well-known and appealingly simple variational quantum Monte Carlo (VMC) method to achieve accuracy well beyond that achieved in previous VMC simulations atoms and small molecules. Using no data other than atomic positions and charges, we predicted the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly-correlated systems, to significantly higher accuracy than the coupled-cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry. This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab-initio quantum chemistry methods.Mr Cassella's PhD project will build on this promising start by using neural network trial wavefunctions to study simple solids. For simplicity, we will start by looking at the uniform electron gas (which will require substantial code development) before moving on to solid hydrogen and perhaps other simple solids. We will also investigate using fermionic neural networks to improve the accuracy of the more sophisticated diffusion quantum Monte Carlo method.