A Straightforward Gradient-Based Approach for High-Tc Superconductor Design: Leveraging Domain Knowledge via Adaptive Constraints

TL;DR

The paper presents KIAGO, a gradient-based, knowledge-integrated framework for designing high-$T_c$ superconductors by directly optimising normalized compositions with two predictors for $T_c$ and formation energy. It eschews training of deep generative models, instead leveraging domain knowledge via masks and a flexible integer-loss to enforce physical constraints, with initialization from promising materials to avoid poor local minima. KIAGO demonstrates superior efficiency and adaptability over elemental substitution and diffusion-based baselines, including the ability to propose novel hydride superconductors beyond the training set. The approach offers a robust, scalable pathway for rapid, constraint-aware materials design in superconductivity and potentially beyond.

Abstract

Materials design aims to discover novel compounds with desired properties. However, prevailing strategies face critical trade-offs. Conventional element-substitution approaches readily and adaptively incorporate various domain knowledge but remain confined to a narrow search space. In contrast, deep generative models efficiently explore vast compositional landscapes, yet they struggle to flexibly integrate domain knowledge. To address these trade-offs, we propose a gradient-based material design framework that combines these strengths, offering both efficiency and adaptability. In our method, chemical compositions are optimised to achieve target properties by using property prediction models and their gradients. In order to seamlessly enforce diverse constraints, including those reflecting domain insights such as oxidation states, discretised compositional ratios, types of elements, and their abundance, we apply masks and employ a special loss function, namely the integer loss. Furthermore, we initialise the optimisation using promising candidates from existing dataset, effectively guiding the search away from unfavourable regions and thus helping to avoid poor solutions. Our approach demonstrates a more efficient exploration of superconductor candidates, uncovering candidate materials with higher critical temperature than conventional element-substitution and generative models. Importantly, it could propose new compositions beyond those found in existing databases, including new hydride superconductors absent from the training dataset but which share compositional similarities with materials found in literature. This synergy of domain knowledge and machine-learning-based scalability provides a robust foundation for rapid, adaptive, and comprehensive materials design for superconductors and beyond.

Figures (7)

• Figure 1: Overview of Knowledge-Integrated Adaptive Gradient-based Optimisation (KIAGO). KIAGO simultaneously maximises $T_c$ and minimises the formation energy by optimising the input composition using two pretrained models and their gradients. Through the use of fixed composition vectors, masks, and specialised loss functions, KIAGO enables flexible control of the composition in three ways: 1. Element control. Specific elements can be fixed and excluded from the optimisation target to perform conditional optimisation. KIAGO is also able to control which elements appear during the optimisation via masks. Here, we fix the composition of Barium and exclude Helium from the optimisation using the mask; 2 Restricting the maximum number of elements. We first rank elements by their abundance in the composition and create a mask to keep only the most abundant ones up to a specified cutoff. All other elements are set to zero, ensuring that the total number of elements never exceeds the chosen limit. In this figure, we select the three most abundant elements to build a mask, which then restricts the final composition to those three elements. 3. Normalising the compositional ratios to small integers. Here, we use the loss function $L_{int}^4$ to guide the normalised composition to a composition consisting of four atoms.
• Figure 2: An overview of the integer loss $L_{int}^{4}$ under the assumption that each unit cell contains four atoms. The numbers shown inside the dashed box represent all possible difference combinations between the integer-compatible set $\lbrace c^4_n\rbrace$ and the composition values. The total loss is obtained by selecting the minimum among these combinations for each element (indicated by the black underline) and summing them.
• Figure 3: Comparison of optimisation results under different initialisation methods. Both approaches employ Adam optimiser kingma2014adam with a learning rate of 0.001. (Left) Initialisation by adding noise to an existing superconductor ($\mathrm{CuLa_2O_4}$). (Right) Random initialisation, in which seven elements are chosen arbitrarily and assigned random compositional values.
• Figure A.1: Periodic table, coloured according to the type of orbital occupied by the outermost electron.
• Figure A.2: Example: Representation of the composition $\mathrm{A_2B_3}$