QPI-DeepONet-MAC：一种可扩展且稳定的物理信息深度算子网络混合经典-量子架构
QPI-DeepONet-MAC: A Scalable and Stable Hybrid Classical-Quantum Architecture for Physics-Informed Deep Operator Networks
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中文总结 AI 辅助
本文提出QPI-DeepONet-MAC混合量子-经典架构，通过乘加耦合集成量子电路于PI-DeepONets，证明其通用逼近能力，推导避免贫瘠高原条件、传感器数量下界及信息正则化方案，实现可扩展稳定的参数化PDE求解。
中文摘要 AI 辅助
针对参数化偏微分方程（PDE）的一般算子学习是人工智能与基于物理的建模交叉领域的一个基本挑战。物理信息深度算子网络（PI-DeepONets）将控制方程纳入学习过程，但在高维情况下面临优化困难。量子扩展提供了额外的表示能力，但其可训练性可能受到贫瘠高原（barren plateaus）的阻碍。在此，我们引入QPI-DeepONet-MAC，一种混合量子-经典架构，通过经典表示与量子期望值的乘加耦合（MAC）将参数化量子电路集成到PI-DeepONets中。我们分析性地证明了所提出的架构保留了通用算子逼近能力。此外，我们推导了量子参数梯度的界限，建立了避免贫瘠高原的明确条件以及可训练性的缩放准则。另外，我们推导了在目标精度下所需传感器数量的下界，直接将采样要求与输入函数的正则性、域维度和PDE动力学增长联系起来。最后，我们提出了一种基于量子相干性和态保真度动力学的信息正则化方案，以防止训练过程中的相干性损失并保持量子结构。综合这些结果，为量子增强的物理信息算子学习奠定了理论基础，并将QPI-DeepONet-MAC定位为求解参数化PDE的可扩展、稳定的框架。
英文摘要
General operator learning for parametric partial differential equations (PDEs) is a fundamental challenge at the intersection of artificial intelligence and physics-based modeling. Physics-informed Deep Operator Networks (PI-DeepONets) incorporate governing equations into learning, but face optimization difficulties in high dimensions. Quantum extensions offer additional representational capacity, yet their trainability can be hindered by barren plateaus. Here, we introduce QPI-DeepONet-MAC, a hybrid quantum-classical architecture that integrates parameterized quantum circuits into PI-DeepONets through a multiplicative-and-additive coupling (MAC) of classical representations and quantum expectation values. We prove analytically that the proposed architecture retains universal operator approximation capabilities. Furthermore, we derive bounds on quantum parameter gradients, establishing explicit conditions to avoid barren plateaus alongside a scaling criterion for trainability. Additionally, we derive a lower bound on the number of sensors required for a target accuracy, directly linking sampling requirements to input function regularity, domain dimension, and PDE dynamical growth. Finally, we propose an informational regularization scheme based on quantum coherence and state fidelity dynamics to prevent coherence loss and preserve quantum structure during training. Together, these results establish a theoretical foundation for quantum-enhanced physics-informed operator learning and position QPI-DeepONet-MAC as a scalable, stable framework for solving parametric PDEs.
发表机构
- Federal University of Santa Maria（圣玛丽亚联邦大学）
- National Laboratory for Scientific Computing (LNCC)（国家科学计算实验室）
机构由 AI 辅助整理，请以论文原文为准。
