AI 中文总结
本文提出新型有序扩散核（ODKs），可近似任意伊藤SDE的无穷小生成元，通过松弛势估计为序推断实现平流与扩散解耦，经合成数据验证能准确恢复多种动力系统属性。
AI 中文摘要
我们提出了有序扩散核（Ordered Diffusion Kernels, ODKs），这是一类可近似任意伊藤随机微分方程（Itô Stochastic Differential Equation, SDE）无穷小生成元的新型局部核。ODKs被设计用于先验动力学信息极少的动力系统采样数据。经典扩散核的拉普拉斯算子近似底流形上的拉普拉斯-贝尔特拉米算子；调整归一化会引入依赖于采样密度的平流项；近期的TMDmap将该归一化推广到任意测度，但代价是平流与扩散耦合。更通用的局部核可学习任意二阶椭圆算子，但需基于已知速度场构建，使得任何分析的第一步都可能是不适定的推断问题。为构建ODK，我们首先将势估计问题松弛为更易处理的数据序推断任务，该任务通过序函数表示。我们证明ODK的拉普拉斯算子收敛到具有状态相关各向同性扩散的梯度流SDE的无穷小生成元，且平流与扩散不耦合。我们提供ODK的多种扩展：通过局部序函数实现任意漂移；通过Strang分裂方案实现各向异性扩散；多序函数；以及自调带宽。此外，我们引入两种利用ODK结构的损失函数以解决非参数推断问题。我们在确定性和随机系统的合成数据上验证该框架，证明其可准确恢复算子、速度场、外曲率以及空间依赖的漂移与扩散系数。
英文摘要
We introduce Ordered Diffusion Kernels (ODKs), a novel class of local kernels that can approximate the infinitesimal generator of an arbitrary Itô Stochastic Differential Equation (SDE). ODKs are designed to be applied to data sampled from dynamical systems where little dynamical information is available a priori. The Laplacian of classical diffusion kernels approximates the Laplace-Beltrami operator on the underlying manifold; adjusting the normalisation introduces an advection term that depends on the sampling density; recently, TMDmap generalised this normalisation to target an arbitrary measure, but at the cost of coupling advection to diffusion. More general local kernels can learn arbitrary second-order elliptic operators but are formulated in terms of known velocity fields --- making the first step of any analysis a potentially ill-posed inference problem. To formulate ODK, we first relax the problem of potential estimation to the more tractable task of inferring an ordering of the data, which we represent through an ordering function. We prove ODK's Laplacian converges to the infinitesimal generator of a gradient-flow SDE with state-dependent isotropic diffusion, without coupling advection and diffusion. We provide various extensions of ODK to: arbitrary drifts via local ordering functions; anisotropic diffusions via a Strang splitting scheme; multiple ordering functions; and self-tuning bandwidths. In addition, we introduce two loss functions which exploit the structure of ODKs to solve a non-parametric inference problem. We validate this framework on synthetic data from deterministic and stochastic systems, demonstrating accurate recovery of operators, velocity fields, extrinsic curvature, and spatially dependent drift and diffusion coefficients.
Comments46 pages, 7 figures, 46 page appendix, related to work presented at ECMTB26
