A Spectral-based Physics-informed Finite Operator Learning for Prediction of Mechanical Behavior of Microstructures

A novel physics-informed operator learning technique based on spectral methods is introduced to model the complex behavior of heterogeneous materials. The Lippmann-Schwinger operator in Fourier space is employed to construct physical constraints with minimal computational overhead, effectively eliminating the need for automatic differentiation. The introduced methodology accelerates the training process by enabling gradient construction on a fixed, finite discretization in Fourier space. Later, the spectral physics-informed finite operator learning (SPiFOL) framework is built based on this discretization and trained to map the arbitrary shape of microstructures to their mechanical responses (strain fields) without relying on labeled data. The training is done by minimizing equilibrium in Fourier space concerning the macroscopic loading condition, which also guarantees the periodicity. SPiFOL, as a physics-informed operator learning method, enables rapid predictions through forward inference after training. To ensure accuracy, we incorporate physical constraints and diversify the training data. However, performance may still degrade for out-of-distribution microstructures. SPiFOL is further enhanced by integrating a Fourier Neural Operator (FNO). Compared to the standard data-driven FNO, SPiFOL shows higher accuracy in predicting stress fields and provides nearly resolution-independent results. Additionally, its zero-shot super-resolution capabilities are explored in heterogeneous domains. Finally, SPiFOL is extended to handle 3D problems and further adapted to finite elasticity, demonstrating the robustness of the framework in handling nonlinear mechanical behavior. The framework shows great potential for efficient and scalable prediction of mechanical responses in complex material systems while also reducing the training time required for training physics-informed neural operators.