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arXiv Machine Learning

QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs

QGPINNs is a PyTorch-based physics-informed neural network framework for solving nonlocal differential equations on quantum graphs. It approximates the solution on each edge with a neural network and uses a unified graph‑based loss to enforce governing equations, initial, boundary, and vertex transmission conditions, including continuity, Kirchhoff‑Neumann, and Dirichlet conditions. The framework supports multi‑order fractional elliptic problems and time‑fractional evolution equations, incorporates graph‑adapted learning strategies such as soft/hard constraints, dynamic loss balancing, Fourier feature embeddings, and a learnable singularity‑capturing feature, and extends to inverse problems for identifying fractional orders and physical parameters from noisy data, as validated on benchmark and real‑world networks such as the IEEE 14‑bus system and an agricultural drainage network.

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