Constitutive modeling based on continuum mechanics theory has been a classical approach for modeling the mechanical responses of materials. However, when constitutive laws are unknown or when defects and/or high degrees of heterogeneity are present, these classical models may become inaccurate. In this work, we propose to use data-driven modeling, which directly utilizes high-fidelity simulation and/or experimental measurements to predict a material's response without using conventional constitutive models. Specifically, the material response is modeled by learning the implicit mappings between loading conditions and the resultant displacement and/or damage fields, with the neural network serving as a surrogate for a solution operator. To model the complex responses due to material heterogeneity and defects, we develop a novel deep neural operator architecture, which we coin as the Implicit Fourier Neural Operator (IFNO). In the IFNO, the increment between layers is modeled as an integral operator to capture the long-range dependencies in the feature space. As the network gets deeper, the limit of IFNO becomes a fixed point equation that yields an implicit neural operator and naturally mimics the displacement/damage fields solving procedure in material modeling problems. We demonstrate the performance of our proposed method for a number of examples, including hyperelastic, anisotropic and brittle materials. As an application, we further employ the proposed approach to learn the material models directly from digital image correlation (DIC) tracking measurements, and show that the learned solution operators substantially outperform the conventional constitutive models in predicting displacement fields.
Diffusion-corrected Autoregressive Fourier Neural Operator for Droplet Evolution Prediction
New Jersey Institute of Technology（新泽西理工学院）；University of Georgia（佐治亚大学）；University at Buffalo（纽约州立大学布法罗分校）；Hanyang University（汉阳大学）；Northeastern University（东北大学）
Neural Fourier Modelling: A Highly Compact Approach to Time-Series Analysis
The University of Auckland（奥克兰大学）；School of Computer Science（计算机科学学院）；Acoustics Vibration Research Centre（声学与振动研究中心）；Mechanical Mechatronics Engineering（机械与机电工程学院）；Strong AI Lab（强人工智能实验室）
Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks
School of Mathematics, Institute for Advanced Study, Princeton（普林斯顿大学数学学院、高等研究院）；Brain Cognition and Brain Disease Institutes of Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院脑认知与脑疾病研究所）
On Optimal Sampling for Learning SDF Using MLPs Equipped with Positional Encoding
University of Hong Kong（香港大学）；University of British Columbia（不列颠哥伦比亚大学）；City University of Hong Kong（香港城市大学）；Zhejiang University（浙江大学）；Texas A&M University（德克萨斯农工大学）
Differentiable Multiphysics Co-Optimization via Implicit Neural Representations: A Transient Hamburger-Cooking Benchmark
Department of Materials Science and Engineering, University of Washington（材料科学与工程系，华盛顿大学）；William E. Boeing Department of Aeronautics & Astronautics, University of Washington（威廉·E·博伊宁航空航天与天体航行系，华盛顿大学）
