Neural Stochastic Differential Equations (Neural SDEs) have emerged as powerful mesh-free generative models for continuous stochastic processes, with critical applications in fields such as finance, physics, and biology. Previous state-of-the-art methods have relied on adversarial training, such as GANs, or on minimizing distance measures between processes using signature kernels. However, GANs suffer from issues like instability, mode collapse, and the need for specialized training techniques, while signature kernel-based methods require solving linear PDEs and backpropagating gradients through the solver, whose computational complexity scales quadratically with the discretization steps. In this paper, we identify a novel class of strictly proper scoring rules for comparing continuous Markov processes. This theoretical finding naturally leads to a novel approach called Finite Dimensional Matching (FDM) for training Neural SDEs. Our method leverages the Markov property of SDEs to provide a computationally efficient training objective. This scoring rule allows us to bypass the computational overhead associated with signature kernels and reduces the training complexity from $O(D^2)$ to $O(D)$ per epoch, where $D$ represents the number of discretization steps of the process. We demonstrate that FDM achieves superior performance, consistently outperforming existing methods in terms of both computational efficiency and generative quality.

翻译：神经随机微分方程（Neural SDEs）已成为连续随机过程强大的无网格生成模型，在金融、物理和生物学等领域具有关键应用。先前最先进的方法依赖于对抗性训练（例如GANs）或使用签名核最小化过程间的距离度量。然而，GANs存在不稳定性、模式崩溃以及需要专门训练技术等问题，而基于签名核的方法则需要求解线性偏微分方程并通过求解器反向传播梯度，其计算复杂度随离散化步数呈二次方增长。本文中，我们提出了一类用于比较连续马尔可夫过程的严格适当评分规则。这一理论发现自然引出了用于训练神经随机微分方程的新方法——有限维匹配（FDM）。我们的方法利用SDEs的马尔可夫性质，提供了一个计算高效的训练目标。该评分规则使我们能够规避与签名核相关的计算开销，并将每个训练周期的计算复杂度从$O(D^2)$降低至$O(D)$，其中$D$表示过程的离散化步数。我们证明，FDM在计算效率和生成质量方面均优于现有方法，展现出卓越的性能。