A multi-fidelity Bayesian neural operator for mechanics of spinodal metamaterial

Cellular metamaterials offer a vast design space for tailoring nonlinear mechanical responses, yet exploring this space with conventional modeling approaches is often infeasible or not scalable. To fully exploit their nonlinear behavior for inverse design, it is essential to learn the full stress-strain response rather than relying on bulk quantities, motivating the use of neural operators for function-to-function mapping. However, data-driven modeling of nonlinear response for metamaterials is severely constrained by the limited availability of costly experimental data. Here, we propose a Bayesian multi-fidelity deep operator network that aggregates abundant low-fidelity finite element simulations with sparse high-fidelity experimental data from in-situ nanomechanical experiments on spinodal metamaterials, enabling heterogeneous information aggregation. A hybrid Bayesian active learning strategy is introduced to select informative samples by jointly maximizing uncertainty and geometric diversity of the microstructure, substantially reducing the cost of 3D nonlinear simulations. This approach adaptively trains the low-fidelity operator, which is then augmented by a high-fidelity Bayesian residual learner. We demonstrate that only 22 strategically selected samples from a design pool of 3000 are sufficient to achieve an 84.1% reduction in MSE compared to the high-fidelity baseline. The framework significantly outperforms single-fidelity baselines, providing superior predictions for full nonlinear stress-strain responses as well as stiffness, strength, and energy absorption. This work provides a robust, data-efficient pathway for the inverse design and constitutive modeling of cellular metamaterials.