结构化神经随机微分方程用于泛函校准
Structured Neural SDEs for Functional Calibration
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中文总结 AI 辅助
本文提出SLiSDE，一种基于结构化线性随机层的神经SDE模型，通过门控堆叠和Girsanov倾斜实现高效并行模拟与稳定重要性采样，在功能校准任务中优于全神经基线。
中文摘要 AI 辅助
神经随机微分方程（Neural SDEs）提供了灵活的连续时间生成模型，但通用的神经漂移和扩散网络在长时间跨度上模拟成本高昂，并且当训练信号是路径泛函而非逐点观测时，可能产生不稳定的梯度。我们引入了SLiSDE，一个由结构化线性随机层构建的神经SDE模型家族。在层级别实现了并行时间模拟，而通过门控流入堆叠恢复了表达能力：前一层路径通过学习的门控调制下一层的潜在流。对于稀有路径主导损失的功能校准任务，我们添加了一个可选的Girsanov倾斜，作为学习的重要性采样器，并带有精确的似然比校正。我们证明了适定性、离散化误差界、测度变换的有效性以及一个普适性结果：门控堆叠的终末分布在平方可积分布空间中是稠密的。在功能校准基准上的实验表明，结构化模型优于全神经SDE基线，同时保持并行时间模拟和稳定的重要性权重。
英文摘要
Neural Stochastic Differential Equations (Neural SDEs) provide flexible continuous-time generative models, but generic neural drift and diffusion networks are costly to simulate on long horizons and can give unstable gradients when the training signal is a path functional rather than a pointwise observation. We introduce SLiSDE, a family of Neural SDE models built from structured linear stochastic layers. Parallel-in-time simulation is obtained at the layer level, while expressivity is recovered by gated in-flow stacking: previous-layer paths modulate the next layer's latent flow through learned gates. For functional calibration tasks in which rare paths dominate the loss, we add an optional Girsanov tilt that acts as a learned importance sampler with an exact likelihood-ratio correction. We prove well-posedness, a discretisation error bound, validity of the change of measure, and a universality result: the terminal laws of the gated stack are dense in the space of square-integrable laws. Experiments on functional calibration benchmarks show that the structured model outperforms fully neural SDE baselines while retaining parallel-time simulation and stable importance weights.
