分支物理信息神经网络用于耦合微分方程组
Forked Physics Informed Neural Networks for Coupled Systems of Differential equations
AI总结：
FPINN通过分支结构和进化正则化损失，有效解决耦合微分方程组中的多目标优化和局部极值问题，应用于量子动力学和多领域建模。
摘要：
求解耦合微分方程组（DEs）是科学计算中的核心问题。尽管物理信息神经网络（PINNs）提供了一种有前途的无网格方法，但其标准架构在处理耦合问题中固有的多目标优化冲突和局部极值陷阱时存在困难。为了解决第一个问题，我们提出了一种分支PINN（FPINN）框架，专门用于耦合微分方程组。FPINN采用共享基础网络和独立分支，隔离梯度路径以稳定训练。我们证明了FPINN在模拟由耦合DEs支配的非马尔可夫开放量子动力学中的有效性，其中多目标冲突和局部极值陷阱常导致进化停滞。为克服第二个挑战，我们引入了进化正则化损失，引导模型远离平凡解并确保物理上有意义的进化。我们证明了FPINN在模拟由耦合DEs支配的非马尔可夫开放量子动力学中的有效性，其中多目标冲突和局部极值陷阱常导致进化停滞。对于自旋-玻色子和XXZ模型，FPINN准确捕捉了非马尔可夫特征，如量子相干复兴和信息回流，显著优于标准PINNs。所提出的FPINN架构提供了一个通用且有效的框架，用于解决出现在广泛领域中的耦合方程组，从经典物理到现代人工智能，包括多体旋转动力学、多资产投资组合优化、化学反应动力学和深度表示学习中的应用。
英文摘要：
Solving coupled systems of differential equations (DEs) is a central problem across scientific computing. While Physics Informed Neural Networks (PINNs) offer a promising, mesh-free approach, their standard architectures struggle with the multi-objective optimization conflicts and local optima traps inherent in coupled problems. To address the first issue, we propose a Forked PINN (FPINN) framework designed for coupled systems of DEs. FPINN employs a shared base network with independent branches, isolating gradient pathways to stabilize training. We demonstrate the effectiveness of FPINN in simulating non-Markovian open quantum dynamics governed by coupled DEs, where multi-objective conflicts and local optima traps often cause evolutionary stagnation. To overcome this second challenge, we incorporate an evolution regularization loss that guides the model away from trivial solutions and ensures physically meaningful evolution. We demonstrate the effectiveness of FPINN in simulating non-Markovian open quantum dynamics governed by coupled DEs, where multi-objective conflicts and local optima traps often cause evolutionary stagnation. For the spin-boson and XXZ models, FPINN accurately captures hallmark non-Markovian features, such as quantum coherence revival and information backflow, significantly outperforming standard PINNs. The proposed FPINN architecture offers a general and effective framework for solving coupled systems of equations, which arise across a broad spectrum from classical physics to modern artificial intelligence, including applications in multi-body rotational dynamics, multi-asset portfolio optimization, chemical reaction kinetics, and deep representation learning.
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