Trainable Embedding Quantum Physics Informed Neural Networks for Solving Nonlinear PDEs

This paper introduces TE-QPINNs, a hybrid quantum-classical method that uses trainable feedforward neural networks as embedding functions in quantum circuits to solve nonlinear PDEs. It achieves superior accuracy compared to classical PINNs with the same parameter count.

S. Berger; N. Hosters; M. Möller· Scientific Reports· 2025· DOI 10.1038/s41598-025-02959-z

The core problem

Solving nonlinear partial differential equations (PDEs) is a fundamental challenge in science and engineering. Classical numerical methods often struggle with high-dimensional problems, and while physics-informed neural networks (PINNs) have emerged as a promising alternative, they can be limited by optimization difficulties. This paper proposes a novel approach: the trainable embedding quantum physics informed neural network (TE-QPINN), which combines quantum machine learning (QML) with PINNs in a hybrid framework. The goal is to leverage the advantages of both classical and quantum computers to create algorithms that can potentially run on noisy intermediate-scale quantum (NISQ) devices. The key innovation is the use of feedforward neural networks (FNN) as problem-agnostic embedding functions, providing greater expressibility than previously introduced embeddings. This allows the method to solve a wide range of problems without requiring a problem-specific ansatz. Additionally, a hybrid backpropagation algorithm is introduced for efficient updates of weights and biases in the FNN embedding functions.

The TE-QPINN architecture integrates a classical FNN with a parameterized quantum circuit (PQC). The FNN acts as a trainable embedding function that maps input coordinates to a quantum state, which is then processed by the PQC. The output is measured to approximate the solution of the PDE. The hybrid backpropagation algorithm enables gradient-based optimization of both classical and quantum parameters. The quantum circuit is designed to be hardware-efficient and compatible with NISQ devices. The method is tested on several nonlinear PDEs: the two-dimensional Poisson equation, Burgers' equation, and the Navier-Stokes equations. The performance is compared against classical PINNs with the same number of parameters. The use of trainable embedding allows the quantum circuit to adapt to the problem at hand, enhancing expressibility and potentially leading to better optimization in high-dimensional parameter spaces.

Introduction

Solving nonlinear partial differential equations (PDEs) is a fundamental challenge in science and engineering. Classical numerical methods often struggle with high-dimensional problems, and while physics-informed neural networks (PINNs) have emerged as a promising alternative, they can be limited by optimization difficulties. This paper proposes a novel approach: the trainable embedding quantum physics informed neural network (TE-QPINN), which combines quantum machine learning (QML) with PINNs in a hybrid framework. The goal is to leverage the advantages of both classical and quantum computers to create algorithms that can potentially run on noisy intermediate-scale quantum (NISQ) devices. The key innovation is the use of feedforward neural networks (FNN) as problem-agnostic embedding functions, providing greater expressibility than previously introduced embeddings. This allows the method to solve a wide range of problems without requiring a problem-specific ansatz. Additionally, a hybrid backpropagation algorithm is introduced for efficient updates of weights and biases in the FNN embedding functions.

The superior performance of TE-QPINNs can be attributed to the increased expressibility provided by the trainable embedding. By using an FNN to map inputs to quantum states, the quantum circuit can represent a wider class of functions, making it more adaptable to various PDEs. The hybrid backpropagation algorithm allows for efficient training, overcoming the barren plateau problem often encountered in quantum machine learning. However, the method is still limited by the capabilities of current NISQ devices, such as noise and limited qubit connectivity. Future work could explore error mitigation techniques and more advanced quantum circuits. The authors suggest that TE-QPINNs could be particularly useful for solving high-dimensional PDEs where classical methods are infeasible. The approach is general and can be extended to other types of differential equations and boundary conditions.

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