提出新方法计算随机系统密度响应，用于生成模型训练与优化。
Divergence-kernel method for linear responses of densities and generative models
- 基于散度核公式推导出参数微分的路径表达式，适用于任意时间尺度。
- 在20维洛伦兹系统上验证，实现高效低内存的生成模型训练。
- 适合需要结构先验和长时动态建模的研究者使用。
我们推导了随机动力系统线性响应的散度核公式，其路径表达式针对边际密度或稳态密度的参数导数，而非平均可观测值。该公式适用于任意时间段内的乘法噪声与参数化噪声，无需双曲性假设。进一步提出了基于蒙特卡洛的线性响应算法。我们构建了一种新生成模型框架DK-SDE，其模型为参数化随机微分方程（SDE），以经验数据分布与SDE边际密度之间的KL散度作为训练目标，并可在长时间跨度下对漂移项和扩散项进行参数化，从而显式融入先验结构知识。优化通过散度核方法实现梯度下降，仅需前向过程，显著降低内存开销。我们在20维洛伦兹系统上展示了该模型的有效性。
原文摘要 · Abstract (English)
We derive the divergence-kernel formula for the linear response of random dynamical systems. Specifically, the pathwise expression is for the parameter-derivative of the marginal or stationary density, not an averaged observable. Our formula works for multiplicative and parameterized noise over any period of time; it does not require hyperbolicity. Then we derive a Monte-Carlo algorithm for linear responses. We develop a new framework of generative models, DK-SDE, where the model is a parameterized SDE, that (1) directly uses the KL divergence between the empirical data distribution and the marginal density of the SDE as the training objective, and (2) accommodates parametrizations in both drift and diffusion over a long time span, allowing prior structural knowledge to be incorporated explicitly. The optimization is done by gradient-descent enabled by the divergence-kernel method, which involves only forward processes and therefore substantially reduces memory cost. We demonstrate the new model on a 20-dimensional Lorenz system.
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