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

如果能够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性原理推导出来。有趣的化学系统的精确波函数是遥不可及的，因为它们通常是NP-难计算的，但可以使用多项式缩放算法找到近似值。这些算法中的许多算法的关键挑战是近似波函数的选择，这必须在效率和精度之间找到平衡。神经网络已经显示出令人印象深刻的权力，作为准确的实用函数逼近器和承诺作为一个紧凑的近似波函数的自旋系统，但在电子结构的问题需要波函数，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架构，即费米神经网络，作为多电子系统的强大近似波函数，并表明这可以与众所周知的和吸引人的简单的变分量子蒙特卡罗（VMC）方法相结合，以实现远远超过以前的VMC模拟原子和小分子所达到的精度。除了原子位置和电荷之外，我们没有使用其他数据，我们预测了氮分子和氢链的解离曲线，这两个具有挑战性的强相关系统，比耦合簇方法的准确性要高得多，耦合簇方法被广泛认为是平衡几何条件下量子化学最准确的可扩展方法。这表明，深度神经网络可以提高变分量子蒙特卡罗的准确性，使其优于其他从头算量子化学方法。卡塞拉先生的博士项目将在这个有希望的开始的基础上，通过使用神经网络试验波函数来研究简单的固体。为了简单起见，我们将从均匀电子气开始（这需要大量的代码开发），然后再讨论固体氢和其他简单固体。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.