利用量子物理信息神经网络学习用于投资组合优化的偏微分方程
Learning PDEs for Portfolio Optimization with Quantum Physics-Informed Neural Networks
AI总结：
本文提出量子物理信息神经网络和量子启发型PINN，通过张量分解多项式实现偏微分方程的高效求解，在投资组合优化问题中展现更高精度和更快收敛速度。
摘要：
偏微分方程（PDEs）在金融数学中，特别是在投资组合优化中扮演着关键角色，使用经典数值或神经网络方法求解始终面临重大挑战。本文研究了量子电路在求解PDEs中的潜力。我们设计了一个参数化量子电路（PQC），用于实现基于张量分解的多项式，当对应的张量秩较小时，将量子资源复杂度从指数级降低到多项式级。在此电路基础上，我们开发了量子物理信息神经网络（QPINN）和量子启发型PINN，两者均保证了PDE解的近似存在，并将该近似表示为包含张量分解的多项式。尽管实验中使用了80倍更少的参数，我们的量子模型在求解Merton投资组合优化问题的PDE时，其精度和收敛速度均优于经典全连接PINN。此外，我们的量子模型进一步优于具有相同归纳偏置的经典PINN，提供了量子诱导改进的实验证据，并突显了通往经典和近期量子PDE求解器的资源高效路径。
英文摘要：
Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics-Informed Neural Network (QPINN) and a Quantum-inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation can be represented as a polynomial that incorporates tensor rank decomposition. Numerical experiments are conducted on the Hamilton--Jacobi--Bellman (HJB) PDE arising from the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. The results show that our quantum models achieve lower losses and approximation errors than a classical fully connected PINN while using substantially fewer trainable parameters. Our quantum models further outperform a classical PINN constructed to share a similar inductive bias, providing experimental evidence of quantum-induced improvement in the tested settings and highlighting a resource-efficient pathway toward classical and near-term quantum solvers for PDEs with exploitable solution structure.
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