用神经算子统一求解多类随机微分方程，效率更高。
One model to solve them all: 2BSDE families via neural operators
- 基于柯尔莫哥洛夫-阿诺德网络构建生成式神经算子
- 可高效逼近无限多类二阶倒向随机微分方程的解算子
- 特定结构下参数量仅需多项式增长，远低于一般情况
我们提出一种温和的生成式神经算子模型，利用柯尔莫哥洛夫-阿诺德网络，在具有随机终止时间的有界欧几里得域上求解无限家族的二阶倒向随机微分方程（2BSDEs）。首个主要结果表明，广泛类别的2BSDE家族对应的解算子可被合适的神经算子模型逼近。随后，我们识别出一类结构化（无限）2BSDE家族，其神经算子近似所需的参数量仅随逆近似率呈多项式增长，而一般情况下神经算子的保证要求指数级增长。
原文摘要 · Abstract (English)
We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.
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。