A Multi-Fidelity Bayesian Neural Operator for Mechanics of Spinodal Metamaterial

Abstract

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 epistemic 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 percent 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.

Figures (8)

• Figure 1: The MF operator learning framework. The LF model acts as a frozen baseline, while the HF residual model utilizes a 3D CNN to map the Signed Distance Field to the simulation-experiment discrepancy.
• Figure 2: Schematic diagrams of design angles and different microstructure. (a) Illustration of cone angle constraint for random vectors. (b)Five typical microstructures generated from different design angles
• Figure 3: (a-b) MSE and R-squared vs samples number in four different schemes during active learning. The hybrid strategy is the proposed scheme, combining two terms; random sampling means adding samples without any bias; while the other two strategies correspond to cases where the hybrid strategy degenerates into only including one term. (c-f) Angles and curves distribution when there are only 22 training samples. The orange lines and dots represent 22 training samples, the blue represent 30 test samples, and the gray lines represent stress–strain curves obtained from all 3000 samples in our pools.
• Figure 4: The LF surrogate model's predicted stress–strain curves for 4 samples. From top to bottom, they are: nearly isotropic, general anisotropic, lamellar, and columnar microstructure.
• Figure 5: (a) Mechanics properties prediction using MF surrogate model and HF surrogate model. Orange dots represent the MF prediction, and green dots represent the HF prediction. (b) Stress prediction of MF, LF and HF surrogate model on 5 unseen samples. The LF model represents using only FEM simulation data to predict the stress–strain responses of unseen microstructure; the HF model represents using only experimental data to do prediction; and the MF model is the comprehensive method of the framework proposed in this work, which combines both LF and HF models