基于路径可观测误差界，提升随机微分方程代理模型预测精度。
Goal-oriented learning of stochastic differential equations using error bounds on path-space observables
- 用路径可观测误差界设计新损失函数，实现目标导向学习。
- 在无界时间域上对首达时间均值的预测误差有理论保证。
- 适合需高精度模拟首达时间或数据分布漂移场景的研究者。
随机微分方程（SDE）作为众多动态系统的基本模型，在量化关键性质时数值模拟成本过高。通过高保真系统数据学习SDE漂移函数的代理模型可提高仿真效率。然而，传统损失函数无法为某些路径相关可观测量（如首次到达时间）提供误差保证。本文引入路径空间可观测量的误差界，并将其作为新型变分损失用于目标导向学习漂移函数。证明该误差界适用于广泛可观测量，包括无界时间域上的平均首达时间。通过利用期望路径泛函的弗雷舍导数公式，推导出该目标损失的解析梯度，保持在随机梯度下降中可计算性。实验表明，基于目标学习的过阻尼朗之万系统代理模型，在预测首达时间统计量方面精度更高，且对数据分布偏移具有更强鲁棒性。
原文摘要 · Abstract (English)
Stochastic differential equations (SDEs), which serve as the governing equations for dynamical systems in a broad range of applications, can become cost-prohibitive for numerical simulation at scales necessary for quantifying key properties. Surrogate models of the drift function of an SDE, learned from data of the high-fidelity system, are routinely used to increase the efficiency of simulation and prediction of properties. However, standard choices of loss function for learning the surrogate model fail to provide error guarantees in certain path-dependent observables, such as transition times. This paper introduces an error bound for path-space observables and employs it as a novel variational loss for the goal-oriented learning of the drift function of a SDE. We show the error bound holds for a broad class of observables, including mean first hitting times on unbounded time domains. We derive an analytical gradient of the goal-oriented loss by leveraging the formula for Fréchet derivatives of expected path functionals, which remains tractable for implementation in stochastic gradient descent schemes. We demonstrate that surrogate models of overdamped Langevin systems developed via goal-oriented learning achieve improved accuracy in predicting the statistics of a first hitting time observable and robustness to distributional shift in the data.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。