Learning PDEs for Portfolio Optimization with Quantum Physics-Informed Neural Networks

TL;DR

This paper introduces quantum physics-informed neural networks for solving PDEs in portfolio optimization, demonstrating higher accuracy and efficiency with fewer parameters compared to classical methods.

Contribution

The authors develop a quantum circuit-based neural network approach that reduces resource complexity and improves PDE solving performance in financial applications.

Findings

Quantum models outperform classical PINNs in accuracy and convergence.

Quantum models use 80 times fewer parameters.

Quantum models achieve better results on the Merton portfolio optimization problem.

Abstract

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 is represented as a polynomial that incorporates tensor rank decomposition. Despite using 80 times fewer parameters in experiments, our quantum models achieve higher accuracy and faster convergence than a classical fully connected PINN when solving the PDE for the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. Our quantum models further outperform a classical PINN constructed to share the same inductive bias, providing experimental evidence of quantum-induced improvement and highlighting a resource-efficient pathway toward classical and near-term quantum PDE solvers.