Self-adaptive physics-informed quantum machine learning for solving differential equations

Abstract Chebyshev polynomials have shown significant promise as an efficient tool for both classical and quantum neural networks to solve linear and nonlinear differential equations (DEs). In this work, we adapt and generalize this framework in a quantum machine learning setting for a variety of problems, including the 2D Poisson’s equation, second-order linear DE, system of DEs, nonlinear Duffing and Riccati equation. In particular, we propose in the quantum setting a modified Self-Adaptive Physics-Informed Neural Network approach, where self-adaptive weights are applied to problems with multi-objective loss functions. We further explore capturing correlations in our loss function using a quantum-correlated measurement, resulting in improved accuracy for initial value problems. We analyse also the use of entangling layers and their impact on the solution accuracy for second-order DEs. The results indicate a promising approach to the near-term evaluation of DEs on quantum devices.