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

• URL: http://arxiv.org/abs/2601.04372v1
• Date: Wed, 07 Jan 2026 20:29:51 GMT
• Title: Solving nonlinear PDEs with Quantum Neural Networks: A variational approach to the Bratu Equation
• Authors: Nikolaos Cheimarios,
• Abstract summary: We present a variational quantum algorithm (VQA) to solve the nonlinear one-dimensional Bratu equation.<n>Trial solution incorporates both classical approximations and boundary-enforcing terms.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 both classical approximations 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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