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

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.