发表机构
ETH Zurich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究将神经跳跃常微分方程扩展到无限维函数空间，基于神经算子方法思想构建框架，开发新近似策略证明收敛性，解决了函数值问题只能通过离散化处理导致信息丢失的问题。
AI 中文摘要
本文研究了神经跳跃常微分方程（Neural Jump ODEs）在无限维函数空间的扩展。具体而言，基础过程\(X\)现在取值于\(L^2(\Xi,\mathbb{R}^{d_X})\)而非\(\mathbb{R}^{d_X}\)，算子神经跳跃常微分方程（Operator NJ - ODE）通过生成条件期望的代表来逼近该过程的最优预测器。NJ - ODE模型是一个在线学习连续时间随机过程最优预测的框架，给定离散、可能不规则且不完整的过去观测值。此前该模型扩展到多种情况，但基础过程限于有限维。本文基于神经算子方法的思想，将NJ - ODE框架扩展到无限维输出过程，并开发新的近似策略证明其收敛性，该策略弱化了假设，也推广了有限维设置下的先前工作。
英文摘要
In this paper, we study the extension of Neural Jump ODEs to infinite-dimensional function spaces. In particular, the underlying process $X$ now takes values in $L^2(Ξ, \mathbb{R}^{d_X})$ instead of $\mathbb{R}^{d_X}$ and the Operator NJ-ODE approximates the optimal predictor of this process by producing a representative of the conditional expectation. The NJ-ODE model is a framework for online learning the optimal prediction of continuous-time stochastic processes, given discrete, possibly irregular and incomplete past observations. In a series of works, this model has been extended to deal with generic path-dependent processes, with observation noise and dependent observations, with long-term predictions, and with input-output systems. However, throughout all of these works, the underlying processes were restricted to be finite-dimensional. In particular, function-valued problems, like yield curve or volatility surface predictions, could only be handled through discretization, which inherently leads to a loss of information. In this work, we build on ideas from Neural Operator methods that allow us to extend the NJ-ODE framework to an infinite-dimensional output process. To prove convergence of the NJ-ODE to the optimal prediction process, we develop a new approximation strategy that also generalizes previous works in the finite-dimensional setting by considerably weakening the assumptions.
Comments25 pages
