Machine learning enables the discovery of 2D Invar and anti-Invar monolayers

Materials demonstrating positive thermal expansion (PTE) or negative thermal expansion (NTE) are quite common, whereas those exhibiting zero thermal expansion (ZTE) are notably scarce. In this work, we identify the mechanical descriptors, namely in-plane tensile stiffness and out-of-plane bending stiffness, that can effectively classify PTE and NTE 2D crystals. By utilizing high throughput calculations and the state-of-the-art symbolic regression method, these descriptors aid in the discovery of ZTE or 2D Invar monolayers with the linear thermal expansion coefficient (LTEC) within  ±2 × 10^-6 K^-1 in the middle range of temperatures. Additionally, the descriptors assist the discovery of large PTE and NTE 2D monolayers with the LTEC larger than  ±15 × 10^-6 K^-1, which are so-called 2D anti-Invar monolayers. Advancing our understanding of materials with exceptionally low or high thermal expansion is of substantial scientific and technological interest, particularly in the development of next-generation electronics at the nanometer or even Ångstrom scale.