AI 中文总结
研究路径积分蒙特卡罗中费米子符号问题，通过概率数值方法将其转为统计推断，开发学习费米子交换循环行为的方法，用蒙特卡罗样本训练模型，引入物理先验，推广线性模型，结合神经网络，开发主动采样方法，应用于基准系统实验取得成果。
AI 中文摘要
从路径积分蒙特卡罗中费米子符号问题的概率数值方法出发，我们将费米子可观测量的算术计算重铸为统计推断问题。我们开发了学习由置换群共轭类划分的费米子交换循环行为的方法，扩展了DuBois等人的工作。利用蒙特卡罗样本训练置换族概率和交换置换能量的模型，直接推断费米子能量。通过引入物理理解作为归纳先验，即使在存在严重符号问题的情况下也能得到准确且有用的拟合。我们用贝叶斯先验推广了线性模型，并以其为基线让长短期记忆神经网络学习剩余多体相关性。还开发了由这些模型驱动的主动重要采样方法以减少可观测量的方差。我们将此框架应用于基准系统的小实验，证明在直接蒙特卡罗采样因符号问题失败的情况下，基于推断的框架能提取稳定能量。
英文摘要
Starting from a \emph{probabilistic numerics} approach to the Fermion sign problem in path integral Monte Carlo, we recast the arithmetic calculation of a Fermionic observable as a statistical inference problem. We develop approaches that learn the behaviour of Fermion exchange cycles binned by the conjugacy class of the permutation group (which we term `permutation family'). This extends the work of DuBois, Brown and Alder\cite{dubois2017overcoming} to inhomogeneous and more complex systems. Monte Carlo samples are used to train models for both the probability of a permutation family and the energy of this set of exchange permutations. The overall Fermionic energy is then directly inferred from these models, without using a direct ratio estimator on the Monte Carlo samples. By imposing physical understanding as inductive priors, we produce accurate and useful fits that remain robust even in regimes with severe sign problems. We generalise the linear (ideal-gas style) models of DuBois et al. with Bayesian priors that enforce the intuitive models of Feynman\cite{Feynman1953A} at their asymptotic limits. These linear models serve as the baseline for a Long Short-Term Memory (LSTM) neural network, which is tasked with learning only the residual many-body \emph{correlations} on top of the physical model. We develop active important sampling methods driven by these models, which direct the Monte Carlo chains toward undersampled permutation regions, to efficiently reduce the variance in the observable. We apply this framework to small experiments on benchmark systems: the spin-polarised uniform electron gas, and electrons in a 2D harmonic confining potential. In both cases we demonstrate that this inference-based framework can extract stable energies in regimes where direct Monte Carlo sampling fails due to the sign problem.
Comments16 pages, 2 figures; prepared for the April 2026 The Sign Problem of Fermions workshop at ECT* Villazzano (Trento), Italy
