Self-Adaptive Physics-Informed Quantum Machine Learning for Solving Differential Equations
This work addresses solving differential equations for computational physics and engineering using quantum machine learning, but it is incremental as it adapts existing classical methods to a quantum setting.
The authors tackled solving differential equations by adapting Chebyshev polynomials and a self-adaptive physics-informed neural network approach in a quantum machine learning setting, achieving improved accuracy for problems like the 2D Poisson's equation and nonlinear equations, with results indicating promise for near-term quantum devices.
Chebyshev polynomials have shown significant promise as an efficient tool for both classical and quantum neural networks to solve linear and nonlinear differential equations. 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 differential equation, system of differential equations, nonlinear Duffing and Riccati equation. In particular, we propose in the quantum setting a modified Self-Adaptive Physics-Informed Neural Network (SAPINN) 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 differential equations. The results indicate a promising approach to the near-term evaluation of differential equations on quantum devices.