首个深度学习求解器，高效处理高维随机积分方程。
A deep solver for backward stochastic Volterra integral equations
- 用单阶段神经网络同时逼近两类解场，避免传统方法的嵌套时间步循环。
- 误差受残差和时间步平方根影响，数值实验验证了该收敛速率。
- 可扩展至500维空间变量，且能统一求解耦合系统，适用于金融与控制领域。
本文提出首个针对后向随机伏特拉积分方程（BSVIEs）及其全耦合前后向变体的深度学习求解器。该方法通过单阶段训练神经网络，同时近似两个解场，避免了经典算法中限制效率的嵌套时间步循环。对于解耦情形，我们证明了非渐近误差界，由后验残差项和时间步的平方根依赖项构成。数值实验与该收敛速率一致，并揭示两大特性：“可扩展性”——从低维到500维空间变量，精度稳定，且利用GPU批处理使实际运行时间几乎恒定；“通用性”——同一方法可处理前向动态依赖于后向解的耦合系统。这些结果为高维、时间不一致的随机控制与量化金融问题提供了实用求解路径。
原文摘要 · Abstract (English)
We present the first deep-learning solver for backward stochastic Volterra integral equations (BSVIEs) and their fully-coupled forward-backward variants. The method trains a neural network to approximate the two solution fields in a single stage, avoiding the use of nested time-stepping cycles that limit classical algorithms. For the decoupled case we prove a non-asymptotic error bound composed of an a posteriori residual plus the familiar square root dependence on the time step. Numerical experiments are consistent with this rate and reveal two key properties: \emph{scalability}, in the sense that accuracy remains stable from low dimension up to 500 spatial variables while GPU batching keeps wall-clock time nearly constant; and \emph{generality}, since the same method handles coupled systems whose forward dynamics depend on the backward solution. These results open practical access to a family of high-dimensional, time-inconsistent problems in stochastic control and quantitative finance.
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