We obtain asymptotically sharp error estimates for the consistency error of
the Target Measure Diffusion map (TMDmap) (Banisch et al. 2020), a variant of
diffusion maps featuring importance sampling and hence allowing input data
drawn from an arbitrary density. The derived error estimates include the bias
error and the variance error. The resulting convergence rates are consistent
with the approximation theory of graph Laplacians. The key novelty of our
results lies in the explicit quantification of all the prefactors on
leading-order terms. We also prove an error estimate for solutions of Dirichlet
BVPs obtained using TMDmap, showing that the solution error is controlled by
consistency error. We use these results to study an important application of
TMDmap in the analysis of rare events in systems governed by overdamped
Langevin dynamics using the framework of transition path theory (TPT). The
cornerstone ingredient of TPT is the solution of the committor problem, a
boundary value problem for the backward Kolmogorov PDE. Remarkably, we find
that the TMDmap algorithm is particularly suited as a meshless solver to the
committor problem due to the cancellation of several error terms in the
prefactor formula. Furthermore, significant improvements in bias and variance
errors occur when using a quasi-uniform sampling density. Our numerical
experiments show that these improvements in accuracy are realizable in practice
when using $\delta$-nets as spatially uniform inputs to the TMDmap algorithm.

通过对 Target Measure Diffusion map（TMDmap）进行错误估计，我们发现它具有重要的抽样能力，能够对任意密度的数据进行输入，控制一致性误差，并在分析由过阻尼朗之万动力学系统中的罕见事件的过渡路径理论框架中，特别适合作为无网格求解器来解决反向 Kolmogorov 偏微分方程的问题，并且在使用准均匀抽样密度时可以显著减小偏差和方差误差。

目标测度扩散映射的尖锐误差估计及在转变度问题中的应用

Sharp error estimates for target measure diffusion maps with  applications to the committor problem

Transition path theory (TPT) is a mathematical framework for quantifying rare
transition events between a pair of selected metastable states $A$ and $B$.
Central to TPT is the committor function, which describes the probability to
hit the metastable state $B$ prior to $A$ from any given starting point of the
phase space. Once the committor is computed, the transition channels and the
transition rate can be readily found. The committor is the solution to the
backward Kolmogorov equation with appropriate boundary conditions. However,
solving it is a challenging task in high dimensions due to the need to mesh a
whole region of the ambient space. In this work, we explore the finite
expression method (FEX, Liang and Yang (2022)) as a tool for computing the
committor. FEX approximates the committor by an algebraic expression involving
a fixed finite number of nonlinear functions and binary arithmetic operations.
The optimal nonlinear functions, the binary operations, and the numerical
coefficients in the expression template are found via reinforcement learning.
The FEX-based committor solver is tested on several high-dimensional benchmark
problems. It gives comparable or better results than neural network-based
solvers. Most importantly, FEX is capable of correctly identifying the
algebraic structure of the solution which allows one to reduce the committor
problem to a low-dimensional one and find the committor with any desired
accuracy.

本文介绍了一种基于有限表达式方法（FEX）的 Committor 函数求解器，并在高维基准问题上进行了测试，并发现该方法能够正确识别解决方案的代数结构，从而使 Committor 问题降到低维，并以任何所需精度找到 Committor。