Learning to sample from complex unnormalized distributions is a fundamental challenge in computational physics and machine learning. While score-based and variational methods have achieved success in continuous domains, extending them to discrete or mixed-variable systems remains difficult due to ill-defined gradients or high variance in estimators. We propose a unified, target-gradient-free generative sampling framework applicable across diverse state spaces. Building on the fact that detailed balance implies the time-reversibility of the equilibrium stochastic process, we enforce this symmetry as a statistical constraint. Specifically, using a prescribed physical transition kernel (such as Metropolis-Hastings), we minimize the Maximum Mean Discrepancy (MMD) between the joint distributions of forward and backward Markov trajectories. Crucially, this training procedure relies solely on energy evaluations via acceptance ratios, circumventing the need for target score functions or continuous relaxations. We demonstrate the versatility of our method on three distinct benchmarks: (1) a continuous multi-modal Gaussian mixture, (2) the discrete high-dimensional Ising model, and (3) a challenging hybrid system coupling discrete indices with continuous dynamics. Experiments show that our framework accurately reproduces thermodynamic observables and captures mode-switching behavior across all regimes, offering a physically grounded and universally applicable alternative for equilibrium sampling.

翻译：学习从复杂的未归一化分布中采样是计算物理学和机器学习中的一个基本挑战。虽然基于分数和变分的方法在连续域中取得了成功，但由于梯度定义不明确或估计器方差过高，将其扩展到离散或混合变量系统仍然困难。我们提出了一种统一的、无需目标梯度的生成采样框架，适用于各种状态空间。基于细致平衡意味着平衡随机过程的时间可逆性这一事实，我们将这种对称性作为统计约束强制执行。具体而言，使用一个预设的物理转移核（如Metropolis-Hastings），我们最小化前向与后向马尔可夫轨迹联合分布之间的最大平均差异（MMD）。至关重要的是，该训练过程仅通过接受率进行能量评估，从而绕过了对目标分数函数或连续松弛的需求。我们在三个不同的基准测试上展示了我们方法的普适性：（1）连续多模态高斯混合模型，（2）离散高维伊辛模型，以及（3）一个将离散索引与连续动力学耦合的具有挑战性的混合系统。实验表明，我们的框架能够准确复现热力学可观测量，并在所有体系中捕捉模态切换行为，为平衡采样提供了一个物理基础坚实且普遍适用的替代方案。