Geometric Quantum Physics Informed Neural Network

TL;DR

This paper introduces GQPINNs, a symmetry-aware quantum neural network framework that incorporates geometric structures of PDEs, leading to improved accuracy and efficiency in solving both linear and nonlinear PDEs.

Contribution

The paper develops a novel symmetry-aware extension of QPINNs, integrating geometric symmetries into quantum circuits to enhance PDE solving performance.

Findings

GQPINNs outperform standard QPINNs and classical PINNs in accuracy.

GQPINNs require fewer trainable parameters for comparable or better results.

Symmetry incorporation improves generalization and efficiency in quantum PDE solvers.

Abstract

Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.