Tong-Rui Liu

Sifan Wang, Tong-Rui Liu, Shyam Sankaran et al.

Heterogeneous materials, crucial in various engineering applications, exhibit complex multiscale behavior, which challenges the effectiveness of traditional computational methods. In this work, we introduce the Micromechanics Transformer ({\em Micrometer}), an artificial intelligence (AI) framework for predicting the mechanical response of heterogeneous materials, bridging the gap between advanced data-driven methods and complex solid mechanics problems. Trained on a large-scale high-resolution dataset of 2D fiber-reinforced composites, Micrometer can achieve state-of-the-art performance in predicting microscale strain fields across a wide range of microstructures, material properties under any loading conditions and We demonstrate the accuracy and computational efficiency of Micrometer through applications in computational homogenization and multiscale modeling, where Micrometer achieves 1\% error in predicting macroscale stress fields while reducing computational time by up to two orders of magnitude compared to conventional numerical solvers. We further showcase the adaptability of the proposed model through transfer learning experiments on new materials with limited data, highlighting its potential to tackle diverse scenarios in mechanical analysis of solid materials. Our work represents a significant step towards AI-driven innovation in computational solid mechanics, addressing the limitations of traditional numerical methods and paving the way for more efficient simulations of heterogeneous materials across various industrial applications.

Sifan Wang, Zehao Dou, Tong-Rui Liu et al.

Recent advances in generative modeling -- particularly diffusion models and flow matching -- have achieved remarkable success in synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. Here, we introduce $\textbf{FunDiff}$, a novel framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generate continuous functions evaluable at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, showing that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results show that our method generates physically consistent samples with high fidelity to the target distribution and exhibits robustness to noisy and low-resolution data. Code and datasets are publicly available at https://github.com/sifanexisted/fundiff.