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A Trainable-Embedding Quantum Physics-Informed Framework for Multi-Species Reaction-Diffusion Systems

Ban Q. Tran, Nahid Binandeh Dehaghani, A. Pedro Aguiar, Rafal Wisniewski, Susan Mengel

TL;DR

This work introduces extended Trainable-Embedding Quantum Physics-Informed Neural Networks (x-TE-QPINNs) to solve multi-species reaction–diffusion PDEs by comparing classical embedding (FNN) and fully quantum embedding (QNN) approaches within a unified physics-informed training framework. The method integrates a trainable embedding network, a hardware-efficient variational quantum circuit, and species-specific readouts, all enforced by a PDE-residual loss, boundary conditions, and initial conditions. Key contributions include a controlled comparison of embedding strategies, a differentiable training pipeline combining parameter-shift quantum gradients with automatic differentiation, and empirical results on 1D and 2D RD systems showing quantum embeddings can match or surpass classical embeddings in certain regimes while offering insights into resource requirements. This work provides architectural guidance for resource-efficient quantum PDE solvers and highlights the trade-offs between embedding design, gradient structure, and quantum hardware constraints in near-term quantum computing contexts.

Abstract

Physics-informed neural networks (PINNs) and hybrid quantum-classical extensions provide a promising framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. In this work, we study embedding strategies for trainable embedding quantum physics-informed neural networks (TE-QPINNs) in the context of nonlinear reaction-diffusion (RD) systems. We introduce an extended TE-QPINN (x-TE-QPINN) architecture that supports both classical and fully quantum embeddings, enabling a controlled comparison between feedforward neural network-based feature maps and parameterized quantum circuit embeddings. The first architecture is the classical embedding feed-forward neural network-based TE-QPINN (FNN-TE-QPINN), while the latter variant is a purely quantum one, referred to as quantum embedding neural network-based TE-QPINN (QNN-TE-QPINN). The proposed framework employs hardware-efficient variational quantum circuits and species-specific readout operators to approximate coupled multi-field dynamics while enforcing governing equations, boundary conditions, and initial conditions through a physics-informed loss function. By isolating the embedding mechanism while keeping the variational ansatz, loss formulation, and optimization procedure fixed, we analyze the impact of embedding design on gradient structure, parameter scaling, and quantum resource requirements. Numerical experiments on one- and two-dimensional RD equations demonstrate that quantum embeddings can replace classical embeddings without degradation in solution accuracy and, in certain regimes, exhibit improved optimization behavior compared to classical PINNs and hybrid quantum models with fixed embeddings. These results provide architectural insight into hybrid quantum PDE solvers and inform the design of resource-efficient quantum physics-informed learning methods.

A Trainable-Embedding Quantum Physics-Informed Framework for Multi-Species Reaction-Diffusion Systems

TL;DR

This work introduces extended Trainable-Embedding Quantum Physics-Informed Neural Networks (x-TE-QPINNs) to solve multi-species reaction–diffusion PDEs by comparing classical embedding (FNN) and fully quantum embedding (QNN) approaches within a unified physics-informed training framework. The method integrates a trainable embedding network, a hardware-efficient variational quantum circuit, and species-specific readouts, all enforced by a PDE-residual loss, boundary conditions, and initial conditions. Key contributions include a controlled comparison of embedding strategies, a differentiable training pipeline combining parameter-shift quantum gradients with automatic differentiation, and empirical results on 1D and 2D RD systems showing quantum embeddings can match or surpass classical embeddings in certain regimes while offering insights into resource requirements. This work provides architectural guidance for resource-efficient quantum PDE solvers and highlights the trade-offs between embedding design, gradient structure, and quantum hardware constraints in near-term quantum computing contexts.

Abstract

Physics-informed neural networks (PINNs) and hybrid quantum-classical extensions provide a promising framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. In this work, we study embedding strategies for trainable embedding quantum physics-informed neural networks (TE-QPINNs) in the context of nonlinear reaction-diffusion (RD) systems. We introduce an extended TE-QPINN (x-TE-QPINN) architecture that supports both classical and fully quantum embeddings, enabling a controlled comparison between feedforward neural network-based feature maps and parameterized quantum circuit embeddings. The first architecture is the classical embedding feed-forward neural network-based TE-QPINN (FNN-TE-QPINN), while the latter variant is a purely quantum one, referred to as quantum embedding neural network-based TE-QPINN (QNN-TE-QPINN). The proposed framework employs hardware-efficient variational quantum circuits and species-specific readout operators to approximate coupled multi-field dynamics while enforcing governing equations, boundary conditions, and initial conditions through a physics-informed loss function. By isolating the embedding mechanism while keeping the variational ansatz, loss formulation, and optimization procedure fixed, we analyze the impact of embedding design on gradient structure, parameter scaling, and quantum resource requirements. Numerical experiments on one- and two-dimensional RD equations demonstrate that quantum embeddings can replace classical embeddings without degradation in solution accuracy and, in certain regimes, exhibit improved optimization behavior compared to classical PINNs and hybrid quantum models with fixed embeddings. These results provide architectural insight into hybrid quantum PDE solvers and inform the design of resource-efficient quantum physics-informed learning methods.
Paper Structure (21 sections, 2 theorems, 26 equations, 7 tables, 1 algorithm)

This paper contains 21 sections, 2 theorems, 26 equations, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reaction-Diffusion Model Formulation
  3. Quantum-Assisted Learning Formulation
  4. Physics-Informed Problem Formulation
  5. Trainable Embedding Network
  6. Variational Quantum Circuit Design
  7. Quantum Readout and Observable Design
  8. Physics-Informed Loss Function
  9. Gradient Structure and Derivative Computation
  10. Computational Cost and Parameter Scaling
  11. Representational Capacity and Potential Quantum Advantages
  12. Experimental Environment and Result Analysis
  13. Experiment Setup
  14. Data Processing
  15. Inference Solutions
  16. ...and 6 more sections

Key Result

Proposition 1

Let $R_A$ and $R_S$ denote the PDE residuals evaluated at a finite interior collocation set $\mathcal{S}_{\mathrm{PDE}}\subset\Omega\times(0,T]$, assumed to be a finite set of collocation points. The PDE loss is Define the residual sensitivities Then the gradient of $\mathcal{L}_{\mathrm{PDE}}$ with respect to the embedding parameters $\theta_{emb}$ is given by

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1: Gradient of the PDE Loss w.r.t. Embedding Parameters
  • proof
  • Proposition 2: Gradient of the PDE Loss w.r.t. Variational Parameters
  • proof
  • Remark 2
  • Remark 3: Unified Gradient Structure
  • Remark 4: On circuit architecture: single vs. species-specific circuits