QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
2026-08-28 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed QGPINNs, a method using neural networks to solve complex equations on networks called quantum graphs. Their approach breaks down the graph into edges, solving each with a neural network while ensuring the solutions fit together smoothly and follow physical rules at the graph's connections. They tested their method on special kinds of equations involving fractional calculus and showed it can also find unknown physical parameters from noisy data. The method works well on both theoretical examples and real-world network problems like power grids and irrigation systems.
Quantum graphsPhysics-informed neural networksNonlocal differential equationsFractional elliptic problemsTime-fractional evolution equationsKirchhoff-Neumann conditionsDirichlet boundary conditionsInverse problemsDynamic loss balancingFourier feature embeddings
Authors
Vaibhav Mehandiratta, Saket Ramchandra
Abstract
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.