Source: arXiv:2608.28589 — Vaibhav Mehandiratta, Saket Ramchandra (2026-08-28)
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-term fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, QGPINNs integrates several graph-adaptive learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and learnable singularity features to capture weakly singular solutions. The framework also naturally extends to inverse problems, including identifying the orders of fractional operators 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 an open agricultural drainage network, demonstrating its accuracy, computational efficiency, and physical consistency.
Key Highlights
- PyTorch-based PINN framework for nonlocal (fractional) differential equations on quantum graphs
- Unified graph-based loss couples edge-wise neural approximations with continuity, Kirchhoff-Neumann, and Dirichlet conditions
- Adaptive training strategies: soft/hard constraints, dynamic loss balancing, Fourier feature embeddings, learnable singularity features
- Forward and inverse problems, including fractional order and parameter identification from noisy data
- Validated on benchmark graphs, the IEEE 14-bus system, and an agricultural drainage network