发表机构
Faculty of Health and Technology, Kristiania University of Applied Sciences; Simula Research Laboratory（健康与技术学院，克里斯蒂安尼亚应用科学大学; 西穆拉研究实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出潜在偏微分方程映射技术，用于有限数据下跨几何的物理信息学习。通过变形梯度处理特定几何的偏微分方程残差等，在解决阿利耶夫 - 潘菲洛夫偏微分方程时效果显著，能降低误差且训练计算成本适中，有助于创建可泛化模型。
AI 中文摘要
在本研究中，我们引入了潜在偏微分方程映射，这是一种广泛适用的物理信息学习技术，旨在利用稀疏训练数据实现高效的几何泛化。潜在偏微分方程映射通过变形梯度将特定于几何的偏微分方程残差和边界条件拉回到预定义的潜在几何中，从而能够自动计算传统物理信息机器学习公式中缺失的几何一致形状梯度。我们使用物理信息神经网络和物理信息深度算子网络证明了潜在偏微分方程映射在解决心脏电生理学的各向异性阿利耶夫 - 潘菲洛夫偏微分方程中的效用。阿利耶夫 - 潘菲洛夫偏微分方程是一个具有挑战性的示例：一个非线性、时间相关的偏微分方程基准，其具有尖锐梯度，使用传统数值求解器捕获成本高昂。为了表示有限数据情况，我们仅使用从二维和三维参数化分布中抽取的十五个几何样本训练网络。虽然仿射和剪切变形参数化的几何有适度改进，但潜在偏微分方程映射在选定几何族上显示出显著优势，平均相对L2误差降低约4 - 6倍。此外，我们的结果表明，在网络训练期间应用潜在偏微分方程映射的计算成本适中，在推理时可忽略不计。总之，我们的研究突出了潜在偏微分方程映射如何促进从有限训练几何集创建可泛化的物理信息机器学习模型。
英文摘要
In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data. Latent PDE mapping pulls back geometry-specific PDE residuals and boundary conditions to a predefined latent geometry via the deformation gradient, thereby enabling the automated calculation of geometry-consistent shape gradients that are missing in conventional physics-informed machine learning formulations. We demonstrate the utility of latent PDE mapping in solving the anisotropic Aliev-Panfilov PDE of cardiac electrophysiology using both physics-informed neural networks and physics-informed deep operator networks. The Aliev-Panfilov PDE serves as a challenging exemplar: a nonlinear, time-dependent PDE benchmark with sharp gradients that are expensive to capture using traditional numerical solvers. To represent the limited data regime, we train the networks using just fifteen geometric samples drawn from parameterized distributions in two and three spatial dimensions. While modest improvements appear for geometries parameterized by affine and shear deformations, latent PDE mapping demonstrates significant benefits on select geometric families, achieving a factor ~4-6 reduction in mean relative L2 error. Furthermore, our results show that the computational cost of applying latent PDE mapping was modest during network training, and negligible at inference. Taken together, our study highlights how latent PDE mapping facilitates the creation of generalizable physics-informed machine learning models from limited sets of training geometries.
