标题： 机器学习邂逅$\mathfrak{su}(n)$李代数：通过精确迹守恒增强量子动力学学习 显示英文标题

标题： Machine learning meets $\mathfrak{su}(n)$ Lie algebra: Enhancing quantum dynamics learning with exact trace conservation

摘要： 机器学习（ML）已成为模拟量子耗散动力学的有前景工具。然而，现有方法在建模约化密度矩阵（RDMs）时常难以强制执行关键物理约束，例如迹守恒。尽管物理学知情神经网络（PINN）旨在解决这些挑战，但它们常常无法实现完全的物理一致性。在这项工作中，我们引入了一种新颖的方法，利用$\mathfrak{su}(n)$李代数将RDMs表示为单位矩阵与$n^2 - 1$的厄米、迹零且正交基算符的组合，其中$n$是系统的维度。通过仅学习与该基相关的系数，我们的框架自然确保了精确的迹守恒，因为基的迹零特性限制了迹贡献仅来自单位矩阵。这消除了损失函数中显式迹守恒惩罚项的需求，简化了优化过程并提高了学习效率。我们在两个基准量子系统上验证了我们的方法：自旋-玻色模型和Fenna-Matthews-Olson复合体。通过比较四种神经网络（NN）架构的性能——纯数据驱动的无物理信息神经网络（PUNN）、$\mathfrak{su}(n)$基于李代数的PUNN（$\mathfrak{su}(n)$-PUNN）、传统PINN以及$\mathfrak{su}(n)$基于李代数的PINN（$\mathfrak{su}(n)$-PINN），我们指出了传统方法的局限性，并展示了我们方法在学习量子耗散动力学中的优越准确性、鲁棒性和效率。 显示英文摘要

摘要： Machine learning (ML) has emerged as a promising tool for simulating quantum dissipative dynamics. However, existing methods often struggle to enforce key physical constraints, such as trace conservation, when modeling reduced density matrices (RDMs). While Physics-Informed Neural Networks (PINN) aim to address these challenges, they frequently fail to achieve full physical consistency. In this work, we introduce a novel approach that leverages the $\mathfrak{su}(n)$ Lie algebra to represent RDMs as a combination of an identity matrix and $n^2 - 1$ Hermitian, traceless, and orthogonal basis operators,where $n$ is the system's dimension. By learning only the coefficients associated with this basis, our framework inherently ensures exact trace conservation, as the traceless nature of the basis restricts the trace contribution solely to the identity matrix. This eliminates the need for explicit trace-preserving penalty terms in the loss function, simplifying optimization and improving learning efficiency. We validate our approach on two benchmark quantum systems: the spin-boson model and the Fenna-Matthews-Olson complex. By comparing the performance of four neural network (NN) architectures -- Purely Data-driven Physics-Uninformed Neural Networks (PUNN), $\mathfrak{su}(n)$ Lie algebra-based PUNN ($\mathfrak{su}(n)$-PUNN), traditional PINN, and $\mathfrak{su}(n)$ Lie algebra-based PINN ($\mathfrak{su}(n)$-PINN) -- we highlight the limitations of conventional methods and demonstrate the superior accuracy, robustness, and efficiency of our approach in learning quantum dissipative dynamics.