Annealing-based approach to solving partial differential equations

TL;DR

The paper develops an annealing-based method to solve PDEs by discretizing them into a system of linear equations and reformulating the resulting generalized eigenvalue problem as a Rayleigh-quotient optimization. It combines a two-stage algorithm—a QUBO-based initial guess computed via an Ising machine and a subsequent iterative descent with adaptive mesh refinement—to achieve arbitrary precision without increasing the binary-variable count. Through simulated annealing on Poisson problems, the authors analyze how the required iterations depend on precision, system size, and problem type, noting subexponential growth with small exponents for several cases and highlighting the impact of problem symmetry. The work suggests potential advantages for large-scale PDEs on Ising machines, though it emphasizes the heuristic nature of annealing and the dependence on problem characteristics for practical performance.

Abstract

Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient. The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables. Investigations using simulated annealing demonstrate how the number of iterations scales with system size and annealing time. Computational performance depends on system size, annealing time, and problem characteristics.

Figures (4)

• Figure 1: The number of iterations required for solving the one-dimensional symmetric Poisson equation \ref{['eq:sym']} as a function of the precision parameter $b$. (a) the initial guess stage. (b) iterative descent stage with $\eta=2^{-b+1}$. (c) iterative descent stage with $\eta=0.1$. The dimensions of solution vectors are $n=9$, $n=19$, and $n=39$. Filled symbols with solid lines correspond to short annealing times, and open symbols with dashed lines correspond to long annealing times. Symbols and shaded ranges represent the average and standard deviation, respectively. Symbol size is proportional to the success rate.
• Figure 2: The number of precision updates required for solving the one-dimensional symmetric Poisson equation \ref{['eq:sym']} as a function of the precision parameter $b$. (a) $b$-dependent mesh reduction rate ($\eta=2^{-b+1}$). (b) Constant mesh reduction rate ($\eta=0.1$). Black dashed lines indicate the theoretical estimate from \ref{['eq:n_upd.0']} for $r=\epsilon_0=10^{-8}$. The dimensions of solution vectors are $n=9$, $n=19$, and $n=39$. Filled symbols with solid lines correspond to short annealing times, and open symbols with dashed lines correspond to long annealing times. Symbols and shaded ranges represent the average and standard deviation, respectively.
• Figure 3: The number of iterations at the iterative descent stage required for solving (a) the one-dimensional asymmetric Poisson equation \ref{['eq:asym']} and (b) the two-dimensional Poisson equation \ref{['eq:2D']} as a function of the precision parameter $b$. The dimensions of solution vectors are $n=9$, $n=19$, and $n=39$ for (a) and $n=3^2$, $n=5^2$, and $n=7^2$ for (b). Symbols and shaded ranges represent the average and standard deviation, respectively. Symbol size is proportional to the success rate.
• Figure 4: The average number of iterations at the iterative descent stage for solving the one-dimensional symmetric, asymmetric, and two-dimensional Poisson equations as a function of system size $n$. Results are shown for $b=5$ and $N_{\rm step}=10^3,\, 10^4$. Symbol size is proportional to the success rate. The dashed line represents $ce^{\alpha n}$ with constants $c$ and $\alpha$.