Learning from multiple quantum chemical methods: Δ-learning, transfer learning, co-kriging, and beyond

Quantum chemistry (QC) has a vast variety of different methods, with more accurate methods being generally slower. This has several consequences: one is that it is easier to generate more data with less accurate methods for training machine learning (ML), whereas the availability of more accurate data is limited. Another consequence is that the databases are rich in data generated with different methods. In addition, some quantum chemical properties such as heats of formation at 298 K and atomization energies at 0 K are related, but the computational cost of their generation and therefore availability is different too. Such data sets with data from different sources are known as multifidelity data, and ML provides tools to learn from them. Here, we discuss such standard tools, transfer learning (TL), and co-kriging, as well as more specialized tools used in QC such as Δ-learning and hierarchical ML as well as methods going beyond them. We will show that Δ-learning and related methods provide an efficient way to improve low-level quantum chemical methods. At the end of the chapter, case studies for performing Δ-learning, hierarchical ML, and TL are provided.