Physics-informed mixture-of-experts framework for extended inverse design of programmable tensile metamaterials

Inverse design of mechanical metamaterials enables programmable nonlinear responses for advanced engineering applications. However, existing inverse-design strategies are often constrained by fixed design spaces and the weak physical consistency of purely data-driven models, limiting their flexibility, accuracy, and generalisation. Here, we develop a physics-informed mixture-of-experts (PI-MoE) framework as an extended inverse-design methodology for tensile metamaterials with programmable nonlinear stress-strain responses through modular assembly of micro-topological units. The PI-MoE framework integrates a hybrid forward predictor that decouples a physics-based structural backbone from neural-network residual learning, together with a generative mixture-of-experts inverse model. Consequently, this hybrid forward predictor achieves high data efficiency, predicting the forward response using only 10% of the training dataset and reducing the prediction error by two orders of magnitude compared to purely data-driven models. For inverse design, it achieves a high coefficient of determination (R2) of 0.991 on the test set. Notably, when the design space is expanded by introducing a high-stiffness unit, the framework maintains a high R2 of 0.984 with only 50 additional finite-element samples, demonstrating its robust generalisation and limited-data adaptation. This study establishes an extensive paradigm for the inverse design of complex mechanical metamaterials, enabling the rapid realisation of structures with unprecedentedly tailored properties.