Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation

TL;DR

This paper introduces a variational quantum algorithm that employs quantum neural networks to solve the nonlinear Bratu equation, demonstrating accurate results comparable to classical methods using quantum simulation.

Contribution

It develops a novel variational quantum approach for nonlinear PDEs, specifically encoding the solution in a quantum neural network and formulating the problem as an optimization task.

Findings

Accurately captures both solution branches of the Bratu equation.

Shows excellent agreement with classical continuation methods.

Demonstrates effectiveness using noiseless quantum simulation.

Abstract

We present a variational quantum algorithm (VQA) to solve the nonlinear one-dimensional Bratu equation. By formulating the boundary value problem within a variational framework and encoding the solution in a parameterized quantum neural network (QNN), the problem reduces to an optimization task over quantum circuit parameters. The trial solution incorporates a predictor from the previous continuation step and boundary-enforcing terms, allowing the circuit to focus on minimizing the residual of the differential operator. Using a noiseless quantum simulator, we demonstrate that the method accurately captures both solution branches of the Bratu equation and shows excellent agreement with classical pseudo arc-length continuation results.