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

Forum topic · 小凯 · 2026-09-01

Summary

Researchers Vaibhav Mehandiratta and Saket Ramchandra present QGPINNs, a PyTorch-based physics-informed neural network (PINN) framework for numerically solving nonlocal differential equations on quantum graphs. Each edge of the graph hosts a neural network approximation, and a unified graph-based loss enforces governing equations along with initial, boundary, and vertex transmission conditions, including continuity, Kirchhoff-Neumann vertex conditions, and Dirichlet boundary conditions. The framework targets two nonlinear model classes: multi-term fractional elliptic problems and time-fractional evolution equations on quantum graphs. Training accuracy and stability are improved via soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and learnable singularity features for weakly singular solutions. QGPINNs also extends to inverse problems such as identifying fractional operator orders and physical parameters from noisy observations. Validation on benchmark graphs and real-world networks, including the IEEE 14-bus system and open agricultural drainage networks, demonstrates accuracy, efficiency, and physical consistency. (arXiv: 2608.28589)

Overview

Field: Machine Learning Authors: Vaibhav Mehandiratta, Saket Ramchandra arXiv: 2608.28589

QGPINNs is a physics-informed neural network framework developed in PyTorch for the numerical solution of nonlocal differential equations on quantum graphs.

Key Ideas

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.
The formulation incorporates standard continuity and Kirchhoff-Neumann vertex conditions, as well as Dirichlet boundary conditions, coupling local edge-wise neural approximations into a global solution on the graph.

Model Classes

The framework is developed for two representative classes of nonlinear models:

Multi-term fractional elliptic problems
Time-fractional evolution equations on quantum graphs

Training Strategies

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

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

Inverse Problems

The framework naturally extends to inverse problems, including identifying fractional operator orders and physical parameters from noisy observation data.

Validation

Numerical experiments on benchmark graph structures and real-world networks—including the IEEE 14-bus system and open agricultural drainage networks—demonstrate the accuracy, computational efficiency, and physical consistency of the proposed framework.

--- *Auto-collected on 2026-09-01*