Self-Adaptive Physics-Informed Quantum Machine Learning for Solving Differential Equations

TL;DR

The paper presents a self-adaptive physics-informed quantum machine learning framework that leverages Chebyshev feature maps to solve diverse differential equations on near-term quantum devices. By coupling a variational quantum circuit with multi-objective loss balancing via trainable SAPINN weights and exploring correlated measurements through tensor-product Pauli-Z observables, the approach achieves improved convergence and accuracy across Riccati, a system of DEs, second-order ODEs, Duffing, and 2D Poisson problems. Key contributions include derivations for second-order derivatives within the quantum Chebyshev framework, the use of mask-based self-adaptive weights, and the demonstration that entangling layers and multi-variable quantum feature maps enhance performance and generalization. This work indicates a promising path for quantum PDE/ODE solvers on near-term devices, with potential extensions to higher dimensions and more complex multiphysics problems, while also highlighting ongoing challenges in optimization and scalability.

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.

Figures (11)

• Figure 1: Quantum circuit for solving differential equations. On the left, a feature map ($\phi(x_i)$) block is used for encoding the value of a function at a specific input $x_i$, followed by a variational parameter block $\hat{\mathcal{U}}_{\theta}$, in which $\theta$ is a set of variational parameters optimized over cost function. In the end, the desired function $(f(x_i))$ in the differential equation is evaluated as an expectation value of the circuit based on a chosen operator $\hat{\mathcal{C}}$.
• Figure 2: Visualization of a quantum circuit for encoding two independent variables using separate feature maps $\hat{\mathcal{U}}_{\phi_x}(x) = \bigotimes_{j=1}^{N_x} R_{Y,j}(\phi_x(x))$ and $\hat{\mathcal{U}}_{\phi_y}(y) = \bigotimes_{m=1}^{N_y} R_{Y,m}(\phi_y(y))$. The qubit numbering for each feature map starts from $1$ as they must be encoded as independent Chebyshev polynomials into the circuit. Also, the variational part of the quantum circuit (in red box) and measurement of the circuit (in green box) are represented following the feature map blocks.
• Figure 3: Solution of Riccati equation \ref{['eq:Problem1_riccati']} and optimizer convergence. (\ref{['fig:Problem1a']}) Visualization of comparison between true solution obtained numerically by the Runge-Kutta method and a classical machine learning approach using PINNs with Adam and L-BFGS optimizer, (\ref{['fig:Problem1b']}) results for the quantum approach using the summation $\hat{\mathcal{C}} = \sum_j^N Z_j$ (referred as Add) and tensor product $\hat{\mathcal{C}} = \bigotimes_{j=1}^{N}Z_j$ (referred as Prod) of Pauli-Z operators for different learning rates (lr) and (\ref{['fig:Problem1c']}) their corresponding losses over the number of optimization iterations.
• Figure 4: Solving of a system of coupled differential equations (refer \ref{['eq:Problem2_system1']}, \ref{['eq:Problem2_system2']}, and \ref{['eq:Problem2_system3']}). Visualization of (\ref{['fig:Problem2a']}) comparison between classical and quantum solution for $u_1(x)$ and $u_2(x)$ as well as (\ref{['fig:Problem2b']}) the loss for the summation and tensor product of Pauli-Z operators cases over the number of optimization iterations.
• Figure 5: Solution of second-order DE (refer \ref{['eq:Problem3_second_order']}) using constant weights for balancing multi-objective loss function. Visualization of (\ref{['fig:Problem3a']}) reference variational quantum circuit (refer \ref{['fig:QC']}) (referred to as RC). It is arranged as RZ, RX, RZ rotational layers followed by CNOT gates in cyclical order. (\ref{['fig:Problem3b']}) Modified variational circuit with added entangling layers in between rotational layers (referred to as AEC). (\ref{['fig:Problem3c']}) Solutions $u(x)$ obtained from three cases and their comparison with the true solution. Case 1 corresponds to employing of RC variational circuit, and balancing weights for multi-objective loss function $\alpha_f = \alpha_b = 1$. Case 2 corresponds to the RC variational circuit, and weights $\alpha_f = 10^{-1}$ and $\alpha_b = 10^{3}$. Case 3 corresponds to the AEC variational circuit, and weights $\alpha_f = 10^{-1}$ and $\alpha_b = 10^{3}$. (\ref{['fig:Problem3d']}) Comparison of accuracy in terms of $1-R^2$ measurement over iterations among three cases, and checking of (\ref{['fig:Problem3e']}) first derivative $\frac{du}{dx}$ and (\ref{['fig:Problem3f']}) second derivative $\frac{d^2u}{dx^2}$ of solution after training.