论文标题
重新思考光谱功能重建的不良性 - 为什么它从根本上困难以及人工神经网络如何帮助
Rethinking the ill-posedness of the spectral function reconstruction -- why is it fundamentally hard and how Artificial Neural Networks can help
论文作者
论文摘要
通过欧几里得相关函数重建强子光谱函数是晶格QCD计算中重要的任务。但是，在källen-lehmann（KL）频谱表示中，观察到重建在实践中是不适合的。与光谱函数中的点数相比，它通常归因于较少的观察点。在本文中，通过解决连续KL卷积的特征值问题，我们在分析上表明，反转的不良性是基本的，即使是连续相关函数也存在。我们讨论了如何介绍监管机构以减轻困境，其中包括作者最近提出的人工神经网络（ANN）表示。 Rev. D 106（2022）L051502]。使用ANN表示的解决方案的唯一性在分析和数值上得到了验证。还展示了使用不同正则化方案的重建光谱函数以及其本征模分解。我们观察到，具有大量特征值的组件可以通过所有方法可靠地重建，而特征值低的组件则需要受监管因子的约束。
Reconstructing hadron spectral functions through Euclidean correlation functions are of the important missions in lattice QCD calculations. However, in a Källen--Lehmann(KL) spectral representation, the reconstruction is observed to be ill-posed in practice. It is usually ascribed to the fewer observation points compared to the number of points in the spectral function. In this paper, by solving the eigenvalue problem of continuous KL convolution, we show analytically that the ill-posedness of the inversion is fundamental and it exists even for continuous correlation functions. We discussed how to introduce regulators to alleviate the predicament, in which include the Artificial Neural Networks(ANNs) representations recently proposed by the Authors in~[Phys. Rev. D 106 (2022) L051502]. The uniqueness of solutions using ANNs representations is manifested analytically and validated numerically. Reconstructed spectral functions using different regularization schemes are also demonstrated, together with their eigen-mode decomposition. We observe that components with large eigenvalues can be reliably reconstructed by all methods, whereas those with low eigenvalues need to be constrained by regulators.
