InvDesFlow-AL: Active Learning-based Workflow for Inverse Design of Functional Materials

TL;DR

InvDesFlow-AL addresses the inverse design problem for functional inorganic materials by coupling an active-learning diffusion generator with multi-objective query-by-committee data selection to steer generation toward thermodynamically stable, synthesizable structures and target functionalities. The approach delivers a CSP RMSE of $0.0423$ Å on MP-20, generates over 1.6 million low-$E_{\text{hull}}$ candidates, and identifies Li$_2$AuH$_6$ with $T_c \approx 140$ K under ambient pressure, supported by phonon and electronic-structure analyses. The framework integrates a FormEGNN formation-energy predictor, DPA-2 relaxation, and SuperconGNN Tc screening within a closed-loop optimization, enabling rapid exploration of chemical space and robust inverse design. These results demonstrate the potential of active-learning-driven diffusion models to accelerate material discovery, expand accessible chemical spaces, and guide experimental synthesis.

Abstract

Developing inverse design methods for functional materials with specific properties is critical to advancing fields like renewable energy, catalysis, energy storage, and carbon capture. Generative models based on diffusion principles can directly produce new materials that meet performance constraints, thereby significantly accelerating the material design process. However, existing methods for generating and predicting crystal structures often remain limited by low success rates. In this work, we propose a novel inverse material design generative framework called InvDesFlow-AL, which is based on active learning strategies. This framework can iteratively optimize the material generation process to gradually guide it towards desired performance characteristics. In terms of crystal structure prediction, the InvDesFlow-AL model achieves an RMSE of 0.0423 Å, representing an 32.96% improvement in performance compared to exsisting generative models. Additionally, InvDesFlow-AL has been successfully validated in the design of low-formation-energy and low-Ehull materials. It can systematically generate materials with progressively lower formation energies while continuously expanding the exploration across diverse chemical spaces. These results fully demonstrate the effectiveness of the proposed active learning-driven generative model in accelerating material discovery and inverse design. To further prove the effectiveness of this method, we took the search for BCS superconductors under ambient pressure as an example explored by InvDesFlow-AL. As a result, we successfully identified Li\(_2\)AuH\(_6\) as a conventional BCS superconductor with an ultra-high transition temperature of 140 K. This discovery provides strong empirical support for the application of inverse design in materials science.

Figures (11)

• Figure 1: Active learning-based workflow for inversing design of materials. (a) Active learning-based diffusion model for designing functional materials. The core of active learning lies in selecting the most valuable data for models to enhance their performance. It primarily involves three strategies: diversity sampling, expected model change, and query-by-committee. (b) Steps of InvDesFlow-AL consist of the following four stages: first, a pre-trained crystal generation model is constructed. Second, this model is fine-tuned on functional materials. Third, the fine-tuned generator is used to generate candidate crystal structures. Finally, a QBCs-based multi-objective function is applied to select the most informative data, which is then used to further fine-tune the generative model. (c) Applications of InvDesFlow-AL. Stable crystal generation, discovery of high-$T_c$ superconductors, crystal structure prediction, identification of ultra-high temperature ceramics, and guidance for experimental synthesis.
• Figure 2: InvDesFlow-AL for the generation of low formation energy materials. (a) InvDesFlow-AL employs a crystal generation model to generate new crystal structures, followed by formation energy prediction using FormEGNN. A lower threshold is applied to retain newly generated materials for fine-tuning the generative model. After five iterations, a progressive decrease in formation energy is observed. (b) InvDesFlow-AL adopts the same strategy to generate materials with low E$_\text{hull}$. Through multiple rounds of generation, a total of 1,610,600 new crystal structures with E$_\text{hull}$ < 50 meV have been obtained, expanding the chemical space exploration to a broader range of atomic species. (c) Crystal structures containing 2, 3, ..., and up to 7 elements generated by InvDesFlow-AL.
• Figure 3: InvDesFlow-AL for discovering novel high-temperature superconducting materials. (a) Comparison of previously reported high-temperature superconductors and newly discovered materials generated by InvDesFlow-AL. The superconducting transition temperatures of these newly discovered materials span a wide range, from the McMillan limit to the liquid nitrogen temperature region. The inset in (a) shows the crystal structure of Li$_2$AuH$_6$. (b) Phonon dispersion and phonon density of states of Li$_2$AuH$_6$, indicating its dynamical stability. (c) Electronic band structure and density of states of Li$_2$AuH$_6$.
• Figure 4: InvDesFlow-AL for generating ultra-high-temperature ceramics. (a)-(c) Synthesized UHTCs (ZrB$_2$, HfC, TiC) existing in the fine-tuning dataset, demonstrating the model’s capacity to reproduce known high-performance ceramics. (d)-(f) Novel UHTCs (TaB$_2$, ZrC, HfN) absent from the training data, whose exceptional properties—high-temperature resistance, oxidation resistance, and thermal shock resilience—have been validated through theoretical/experimental studies. This systematic validation underscores InvDesFlow-AL’s capability to design unreported, high-stability ceramics beyond existing databases.
• Figure 5: SuperconGNN model architecture and performance. (a) The input layer, Embedding layer, encoding layer, and prediction layer of the crystal graph in SuperconGNN. (b) Performance comparison of different models in predicting high-temperature superconducting materials. (c) Accuracy of AI predictions: whether AI can correctly classify materials with superconducting transition temperatures above 5K, 10K, ..., up to 60K into their corresponding intervals.