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This research introduces QGPINNs, a specialized physics-informed neural network framework designed to solve nonlocal differential equations on quantum graphs with high accuracy. By integrating graph-adapted learning strategies such as dynamic loss balancing and singularity-capturing features, the method effectively couples local edge-wise approximations into global solutions for complex fractional models. The framework demonstrates robust performance in both forward simulations and inverse problems, including parameter identification from noisy data on real-world network structures.
QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
arXiv:2608.28589v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2608.28589 [cs.LG]
(or arXiv:2608.28589v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.28589 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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