Gradient-free optimization of non-differentiable hybrid neural solvers for spatially heterogeneous composites

• Gradient-free training of hybrid neural models using mini-batched EKI algorithm. • Efficient learning of constitutive laws for fiber-reinforced composite materials. • Neural network integrated with FEM predicts deformations using indirect displacement data. • Demonstrates robustness under unseen material distributions and varying loading conditions. The data-driven machine learning paradigm typically requires high-quality, large-scale datasets for training neural networks, which are often unavailable in many scientific and engineering applications. Integrating physics equations into machine learning models, either fully or partially, can mitigate these data requirements and improve generalizability; however, such approaches frequently rely on differentiable programming frameworks. This ability poses significant challenges when legacy or commercial numerical solvers, which are often non-differentiable and difficult to modify without introducing code changes, are integrated. This work addresses these challenges by leveraging the mini-batching iterative ensemble Kalman inversion (EKI) algorithm as a gradient-free training framework for hybrid neural models. The use of stochastic mini-batching significantly enhances the computational efficiency and convergence of EKI, making it well-suited for high-dimensional learning problems. The proposed method is demonstrated for modeling a fiber-reinforced composite plate, where heterogeneous local constitutive laws are parameterized by a trainable neural network embedded within the FEniCS finite element solver. Using the displacement field as indirect data, the hybrid neural FEM solver successfully predicts deformations by learning the local constitutive laws, even for unseen fiber volume fraction distributions and varying test loading conditions. These results demonstrate the effectiveness of iterative EKI in training hybrid neural models with non-differentiable components, paving the way for broader adoption of hybrid neural models in scientific and engineering applications.