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arXiv · 2601.04372

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

Nikolaos Cheimarios
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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.

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Nikolaos Cheimarios. 2026-01-07. Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation. https://arxiv.org/abs/2601.04372

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