Accurate modeling of quantum many-body systems often requires computationally expensive simulations such as Density Matrix Renormalization Group (DMRG) or Quantum Monte Carlo (QMC) calculations. These methods, while precise, impose significant time and resource constraints, limiting their use in exhaustive parameter exploration. Moreover, these expensive simulations can contain variable errors over the large unknown parameter space, which needs to be quantified and propagated. Thus, predictive modelling is required to estimate the functional space accurately over scarcely sampled data with heteroskedastic noise, while preserving the physical relevance of the estimation. Therefore, we present a Physically Constrained Ensemble Gaussian Process (pc-EGP) framework designed to efficiently model complex and noisy quantum systems under physical consistency constraints. The proposed method first enforces physical constraints as a user controlled weighted penalty to the data-driven loss function of the Gaussian Process (GP) surrogates. Then an ensemble of such GP models is trained with variable noisy simulations via numerical quadrature method where these multiple GP(s) at different nodes is integrated as a quadrature weighted average. We first demonstrate the framework on synthetically generated data before applying to quantum systems. In the first case study, we leverage DMRG simulations of the Bose-Hubbard Model to predict the critical interaction parameter Uc governing the superfluid-to-Mott-insulator transition. In the second case study, we demonstrate our method on QMC simulations, of a quantum liquid confined inside a nanoporous silicate with the goal of optimizing a chemical environment to realize a one-dimensional superfluid. Compared to conventional GP, pc-EGP achieves a better balance of accuracy and physically meaningful predictions.

翻译：量子多体系统的精确建模通常需要高计算成本的模拟，如密度矩阵重正化群（DMRG）或量子蒙特卡洛（QMC）计算。这些方法虽精度高，但受限于时间和资源约束，难以进行全面的参数空间探索。此外，在未知的大参数空间中，这些昂贵模拟可能包含可变误差，需对其量化与传播。因此，需要构建预测模型，在保留物理相关性的前提下，基于含异方差噪声的稀疏采样数据精确估计函数空间。为此，我们提出物理约束集成高斯过程（pc-EGP）框架，旨在高效建模物理一致性约束下的复杂含噪量子系统。该方法首先将物理约束作为用户控制的加权惩罚项，施加于高斯过程（GP）替代模型的数据驱动损失函数；随后通过数值求积方法训练此类GP模型的集成，将不同节点处的多个GP以求积加权平均值形式融合。我们在合成数据上验证该框架后，将其应用于量子系统。首个案例利用Bose-Hubbard模型的DMRG模拟，预测控制超流-莫特绝缘体转变的临界相互作用参数Uc；第二个案例则基于纳米孔硅酸盐中受限量子液体的QMC模拟，旨在优化化学环境以实现一维超流。与常规GP相比，pc-EGP在预测精度与物理意义之间实现了更优平衡。