Zhang Zhizhou, Lee Jeong-Ho, Sun Lingfeng, Gu Grace X
Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA 94720, USA.
PNAS Nexus. 2024 May 8;3(5):pgae186. doi: 10.1093/pnasnexus/pgae186. eCollection 2024 May.
Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This article also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pretrained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pretraining.
偏微分方程（PDEs）的数值解对于需要进行大量数据筛选的材料结构设计至关重要。然而，传统的数值方法需要大量的计算资源，这凸显了对创新优化算法的需求，以简化设计探索。基于直接梯度的优化算法虽然有效，但依赖于设计初始化，并且需要复杂的、特定于问题的灵敏度推导。机器学习的出现为处理大参数空间提供了一个有前途的替代方案。为了进一步减轻数据依赖性，研究人员开发了物理信息神经网络（PINNs），以便直接从偏微分方程中学习。然而，PINNs的内在连续性要求限制了它们在结构力学问题中的应用，特别是对于复合材料。我们的工作通过在物理信息训练过程中用弱形式代替偏微分方程残差来解决这个不连续性问题。所提出的方法在数字材料建模中得到了例证，数字材料是具有极端结构不连续性的复杂复合材料的数学表示。本文还介绍了一个将物理信息损失与设计目标相结合的交互式过程，无需预先训练的代理模型或解析灵敏度推导。结果表明，我们的方法不仅可以在无数据材料代理建模中保持物理准确性，还可以在无需模型预训练的情况下加速直接优化过程。
