用神经网络模拟正倒向随机微分方程，给出误差上界并优化算法框架。
Convergence proofs and strong error bounds for forward-backward stochastic differential equations using neural network simulations
- 结合神经网络与多级蒙特卡洛，构建求解框架。
- 误差上界由离散误差和网络近似误差的最大值决定。
- 揭示现有损失函数的偏差与方差问题，适合数值方法研究者。
我们引入正倒向随机微分方程，阐明其解与偏微分方程解之间的联系，该联系由费曼-卡茨定理建立。回顾利用神经网络近似高维偏微分方程解的方法，以及通过多级蒙特卡洛近似随机微分方程解的技术。将多级蒙特卡洛方法与神经网络框架相结合，基于E等人的设定及Raissi的工作，提出新的数值分析，为Raissi框架提供强误差界。结果表明，总强误差由离散化误差与神经网络近似误差的最大值决定。分析对多级蒙特卡洛应用至关重要，我们提出合适框架以利用所揭示的多级估计器方差结构。此外，针对Raissi倡导的损失函数，暴露其局限性，量化其偏差与方差。最后，提出若干未来研究方向，预期可带来显著洞察与加速效果。
原文摘要 · Abstract (English)
We introduce forward-backward stochastic differential equations, highlighting the connection between solutions of these and solutions of partial differential equations, related by the Feynman-Kac theorem. We review the technique of approximating solutions to high dimensional partial differential equations using neural networks, and similarly approximating solutions of stochastic differential equations using multilevel Monte Carlo. Connecting the multilevel Monte Carlo method with the neural network framework using the setup established by E et al. and Raissi, we provide novel numerical analyses to produce strong error bounds for the specific framework of Raissi. Our results bound the overall strong error in terms of the maximum of the discretisation error and the neural network's approximation error. Our analyses are necessary for applications of multilevel Monte Carlo, for which we propose suitable frameworks to exploit the variance structures of the multilevel estimators we elucidate. Also, focusing on the loss function advocated by Raissi, we expose the limitations of this, highlighting and quantifying its bias and variance. Lastly, we propose various avenues of further research which we anticipate should offer significant insight and speed improvements.
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