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

Forum topic · 小凯 · 2026-09-01

Summary

QGPINNs is a PyTorch-based physics-informed neural network (PINN) framework for numerically solving nonlocal differential equations on quantum graphs, presented in arXiv paper 2608.28589 by Vaibhav Mehandiratta and Saket Ramchandra. In the framework, 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—including standard continuity, Kirchhoff-Neumann vertex conditions, and Dirichlet boundary conditions—coupling local edge-wise approximations into a global solution. The framework targets two classes of nonlinear models: multi-order fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, it incorporates graph-adaptive learning strategies such as soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and learnable singularity features for weakly singular solutions. QGPINNs also extends naturally to inverse problems, including identification of fractional operator orders and physical parameters from noisy observational data. Validation via numerical experiments on benchmark graph structures and real-world networks (IEEE 14-bus system and an open agricultural drainage network) demonstrates accuracy, computational efficiency, and physical consistency.

Overview

Field: Machine Learning Authors: Vaibhav Mehandiratta, Saket Ramchandra Published: 2026-08-28 arXiv: 2608.28589

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
Time-fractional evolution equations on quantum graphs

To improve accuracy and training stability, QGPINNs integrates several graph-adaptive learning strategies:

Soft and hard constraint enforcement
Dynamic loss balancing
Fourier feature embeddings
Learnable singularity features to capture weakly singular solutions

The framework also naturally extends to inverse problems, including identification of fractional operator orders and physical parameters from noisy observational data.

Validation

The authors validate the proposed framework 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.

--- *Source: arXiv:2608.28589*