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. 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.

References in corpus (13)

• Barren plateaus in quantum neural network training landscapes
• Quantum Circuit Learning
• Quantum principal component analysis
• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• Variational quantum algorithms for nonlinear problems
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
• Solving nonlinear differential equations with differentiable quantum circuits
• Avoiding barren plateaus via transferability of smooth solutions in Hamiltonian Variational Ansatz
• Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
• Quantum Kernel Methods for Solving Differential Equations
• A Hybrid Quantum-Classical CFD Methodology with Benchmark HHL Solutions
• Quantum Chebyshev Transform: Mapping, Embedding, Learning and Sampling Distributions

Cited by in corpus (2)

• Resource-Efficient Hadamard Test Tailored Variational Framework for Nonlinear Dynamics on Quantum Computers
• Vortex Detection from Quantum Data