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-term fractional elliptic problems and time-fractional evolution equations on quantum graphs. To enhance accuracy and training stability, QGPINNs integrates multiple graph-adaptive learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and learnable singularity features to capture the weakly singular solutions arising in these problems.
The framework also extends naturally to inverse problems, including the identification of fractional operator orders and physical parameters from noisy observational data. The proposed framework is validated through numerical experiments on benchmark graph structures and real-world networks, including the IEEE 14-bus system and open agricultural drainage networks, demonstrating its accuracy, computational efficiency, and physical consistency.
Paper link: arXiv:2608.28589