QGPINNs：用于量子图上非局部微分方程的物理信息神经网络框架
QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
- Birla Institute of Technology and Science, Pilani（贝拉理工科学技术大学皮尔尼分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究针对量子图上非局部微分方程的数值求解，提出基于PyTorch的QGPINNs框架，整合图适配学习策略，可求解两类非线性模型，还可扩展至逆问题，经基准与真实网络实验验证有效。
AI中文摘要：
我们提出了QGPINNs，这是一个基于PyTorch开发的物理信息神经网络框架，用于数值求解量子图上的非局部微分方程。该框架被设计为通用计算实现，其中图的每条边上的解由神经网络近似，同时统一的基于图的损失函数会强制满足控制方程以及初始、边界和顶点传输条件。特别地，该公式将标准连续性、基尔霍夫-诺伊曼顶点条件和狄利克雷边界条件纳入学习过程，以将逐边的局部神经近似耦合为图上的全局解。该框架针对两类代表性非线性模型开发：量子图上的多阶分数椭圆问题和时间分数演化方程。为提高精度和训练稳定性，QGPINNs整合了多种适配图的学习策略，包括软约束与硬约束执行、动态损失平衡、傅里叶特征嵌入，以及针对所考虑问题中出现的弱奇异解的可学习奇异捕获特征。该框架还可自然扩展至逆问题，包括从带噪观测数据中识别分数算子的阶数和物理参数。我们通过在基准图结构和真实世界网络上的数值实验验证所提框架的精度、计算效率和物理一致性，这些网络包括IEEE 14节点系统和明渠农业排水网络。
英文摘要：
We propose QGPINNs, a physics-informed neural network framework developed in PyTorch for the numerical solution of nonlocal differential equations on quantum graphs. The framework is designed as a general computational implementation in which the solution on each edge of the graph is approximated by a neural network, while a unified graph-based loss function enforces the governing equations together with initial, boundary, and vertex transmission conditions. In particular, the formulation incorporates standard continuity and Kirchhoff-Neumann vertex conditions and Dirichlet boundary conditions into the learning process to couple the local edge-wise neural approximations into a global solution on the graph. The framework is developed for two representative classes of nonlinear models: multi-order fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, QGPINNs integrates several graph-adapted learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and a learnable singularity-capturing feature for weakly singular solutions arising in the considered problems. The framework also extends naturally to inverse problems, including the identification of the orders of fractional operators and physical parameters from noisy observational data. We validate the accuracy, computational efficiency, and physical consistency of the proposed framework through numerical experiments on benchmark graph structures and real-world networks, including the IEEE 14-bus system and an open-channel agricultural drainage network.
