优化参数化物理信息神经网络以求解多层静态线性弹性偏微分方程
Optimizing Parameterized Physics-Informed Neural Networks to Solve Multilayered Static Linear Elastic PDEs
AI总结：
该研究针对传统FEM计算成本高的问题，提出P2INNs框架求解多层静态线性弹性PDE，其误差满足设计需求，可替代传统FEM并加速防护结构层状架构的逆向优化。
AI中文摘要：
设计用于可控变形的多层材料，严重受限于传统工业级有限元方法（FEM）极高的计算成本与耗时，这种过度开销极大限制了设计空间探索及基于梯度的优化。为简化工作流程，我们提出了参数化物理信息神经网络（P2INNs）框架，该框架包含三个核心组件：一是采用Hex8单元与三线性基函数的FEM基线；二是适用于可变材料刚度与层厚的P2INN位移场模型；三是基于FEM的参考评估。PINN通过带有优先考虑物理特性的训练策略的Navier-Cauchy残差来强制实现静态线性弹性，这些策略包括分层偏微分方程分解、界面连续性惩罚以及顺应性感知缩放。对于单层纯物理驱动配置，PINN与FEM基准相比实现了1.56%的平均体积平均绝对误差（MAE）和2.87%的最坏情况体积MAE；对于带有受控监督训练的三层配置，PINN实现了2.53%的平均体积MAE和4.66%的最坏情况体积MAE，达到了初步设计空间探索所需的近5%最坏情况体积MAE目标。训练后的P2INN模型仍为轻量级模型，可通过简单前向传播进行后续计算，成为传统FEM的替代方案。通过大幅减少对昂贵FEM评估的需求，该方法可加速各类重载或潜在碰撞领域中防护结构的先进层状架构的逆向设计与优化。
英文摘要：
Designing multilayered materials for controlled deformation is heavily bottlenecked by the immense computational costs and time of traditional, industrial-grade finite element methods (FEM). This excessive expense severely limits design space exploration and gradient-based optimization. To streamline the workflow, we propose a framework for parameterized physics-informed neural networks (P2INNs). The framework encompasses three core components: I. a FEM baseline with a Hex8 element and trilinear basis functions, II. a P2INN displacement field model across variable material stiffness and layer thicknesses, and III. FEM-referenced evaluation. The PINN enforces static linear elasticity via Navier-Cauchy residuals with training strategies that prioritize physics, including layerwise PDE decomposition, interface continuity penalty, and compliance aware scaling. For one-layer, pure physics-driven configurations, the PINN achieves a mean volume MAE of 1.56% and worst-case volume MAE of 2.87% against the FEM benchmark. For three-layer configurations with controlled supervised training, the PINN achieves a mean volume MAE of 2.53% and worst-case volume MAE of 4.66%, meeting the near-5% worst-case volume-MAE target for preliminary design-space exploration. The trained P2INN model remains a lightweight model with a simple forward pass for future calculations, creating an alternative to traditional FEM. By drastically reducing the need for expensive FEM evaluations, this approach could accelerate the inverse design and optimization of advanced layered architectures for protective structures in various load-heavy or potentially collision-heavy fields.
