Department of Chemistry, University of California, Irvine, California 92697, United States.
Google Research, Mountain View, California 94043, United States.
Acc Chem Res. 2021 Feb 16;54(4):818-826. doi: 10.1021/acs.accounts.0c00742. Epub 2021 Feb 3.
Density functional theory (DFT) calculations are used in over 40,000 scientific papers each year, in chemistry, materials science, and far beyond. DFT is extremely useful because it is computationally much less expensive than electronic structure methods and allows systems of considerably larger size to be treated. However, the accuracy of any Kohn-Sham DFT calculation is limited by the approximation chosen for the exchange-correlation (XC) energy. For more than half a century, humans have developed the art of such approximations, using general principles, empirical data, or a combination of both, typically yielding useful results, but with errors well above the chemical accuracy limit (1 kcal/mol). Over the last 15 years, machine learning (ML) has made major breakthroughs in many applications and is now being applied to electronic structure calculations. This recent rise of ML begs the question: Can ML propose or improve density functional approximations? Success could greatly enhance the accuracy and usefulness of DFT calculations without increasing the cost.In this work, we detail efforts in this direction, beginning with an elementary proof of principle from 2012, namely, finding the kinetic energy of several Fermions in a box using kernel ridge regression. This is an example of orbital-free DFT, for which a successful general-purpose scheme could make even DFT calculations run much faster. We trace the development of that work to state-of-the-art molecular dynamics simulations of resorcinol with chemical accuracy. By training on examples, one bypasses the need to find the XC functional explicitly. We also discuss how the exchange-correlation energy itself can be modeled with such methods, especially for strongly correlated materials. Finally, we show how deep neural networks with differentiable programming can be used to construct accurate density functionals from very few data points by using the Kohn-Sham equations themselves as a regularizer. All these cases show that ML can create approximations of greater accuracy than humans, and is capable of finding approximations that can deal with difficult cases such as strong correlation. However, such ML-designed functionals have not been implemented in standard codes because of one last great challenge: generalization. We discuss how effortlessly human-designed functionals can be applied to a wide range of situations, and how difficult that is for ML.
密度泛函理论（DFT）计算每年在化学、材料科学等领域的超过 40000 篇科学论文中得到应用。DFT 非常有用，因为它的计算成本比电子结构方法低得多，并且可以处理更大规模的系统。然而，任何 Kohn-Sham DFT 计算的准确性都受到所选择的交换相关（XC）能量近似的限制。半个多世纪以来，人类一直在发展这种近似的艺术，使用一般原则、经验数据或两者的结合，通常会得到有用的结果，但误差远远超过化学精度极限（1 kcal/mol）。在过去的 15 年中，机器学习（ML）在许多应用中取得了重大突破，现在正在被应用于电子结构计算。ML 的最近崛起引发了一个问题：ML 能否提出或改进密度泛函近似？成功的话可以极大地提高 DFT 计算的准确性和实用性，而不会增加成本。在这项工作中，我们详细介绍了朝这个方向所做的努力，从 2012 年的一个基本原理证明开始，即使用核岭回归来找到盒子中几个费米子的动能。这是无轨道 DFT 的一个例子，如果成功的通用方案可以使 DFT 计算速度大大加快。我们追踪了这项工作的发展，实现了对间苯二酚的具有化学精度的最先进的分子动力学模拟。通过在示例上进行训练，可以避免显式地找到 XC 泛函。我们还讨论了如何使用这种方法对交换相关能量本身进行建模，特别是对于强关联材料。最后，我们展示了如何通过使用 Kohn-Sham 方程本身作为正则化器，使用具有可微编程的深度神经网络从很少的数据点构建准确的密度泛函。所有这些例子都表明，ML 可以创建比人类更准确的近似，并且能够找到可以处理强关联等困难情况的近似。然而，由于最后一个巨大的挑战——泛化，这样的 ML 设计的泛函还没有被应用到标准代码中。我们讨论了人类设计的泛函如何轻松地应用于广泛的情况，以及 ML 做到这一点有多困难。
