通过图超网络摊销物理信息神经求解器
Amortizing Physics-Informed Neural Solvers via Graph Hypernetworks
- Arizona State University（亚利桑那州立大学）
- Applied Materials Inc.（应用材料公司）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出用图超网络将方程关系显式编码，为相关PDE生成PINN初始化，在固定适应预算下提升求解精度，尤其在双向耦合系统中显著优于系数向量。
AI中文摘要：
在相关偏微分方程之间摊销物理信息神经网络（PINN）需要将每个方程描述给一个可复用的求解器。系数向量在预定义的槽位中编码数值参数，使得算子分配和跨场分配隐含其中。我们在算子图中明确这些关系，其中节点表示场、导数、项和残差，系数保留为项属性。一个图超网络生成对角编码，为每个目标方程初始化一个元训练过的因子化PINN。元训练和目标特定适应使用控制方程和规定条件，无需解标签。我们在固定的适应预算内，通过解精度比较系数向量、基于DeepSets的项集和图条件化。在标量对流-扩散-反应问题中，两种基于项的描述符都提高了高反应精度，且性能相似。在两场Fisher-KPP中，元训练看到非耦合和单向系统；在未见过的双向耦合上进行3,000步适应后，图的平均最终误差比项集低35.7%，比系数向量低67.7%。在固定结构的电容耦合等离子体模型中，系数向量表现最佳。这些结果支持用显式方程关系扩展系数条件化，以进行基于物理的求解器适应。
英文摘要：
Amortizing physics-informed neural networks (PINNs) across related PDEs requires describing each equation to a reusable solver. Coefficient vectors encode numerical parameters in predefined slots, leaving operator and cross-field assignments implicit. We make these relationships explicit in an operator graph, with nodes for fields, derivatives, terms, and residuals and coefficients retained as term attributes. A graph hypernetwork generates diagonal codes that initialize a meta-trained factorized PINN for each target equation. Meta-training and target-specific adaptation use governing equations and prescribed conditions without solution labels. We compare coefficient-vector, DeepSets-based term-set, and graph conditioning by solution accuracy within a fixed adaptation budget. In scalar convection-diffusion-reaction problems, both term-based descriptors improve high-reaction accuracy, with similar performance. In two-field Fisher-KPP, meta-training sees uncoupled and one-way systems; after 3,000 adaptation steps on unseen two-way coupling, the graph's mean final error is 35.7% below the term set and 67.7% below the coefficient vector. In a fixed-structure capacitively coupled plasma model, the coefficient vector performs best. These results support extending coefficient conditioning with explicit equation relationships for physics-based solver adaptation.
