Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While Lévy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.

翻译：极端事件与重尾现象的建模是构建金融、气候科学及安全关键 AI 等可靠预测系统的核心。尽管 Lévy 过程为捕捉跳跃性与重尾特性提供了天然的数学框架，但针对 Lévy 驱动随机微分方程 (SDE) 的贝叶斯推断仍面临现有方法难以解决的挑战：蒙特卡洛方法虽严谨但缺乏可扩展性，而神经变分推断方法虽高效却依赖无法刻画不连续性的高斯假设。为调和这一矛盾，我们提出一种面向 Lévy 驱动 SDE 变分推断的神经指数倾斜框架。该方法通过神经网络对 Lévy 测度进行指数重加权，构建出灵活变分族。该参数化在保持计算可处理性的同时，完整保留了底层过程的跳跃结构。为实现高效推断，我们开发了能生成倾斜测度闭式归一化常数的二次神经参数化、适用于稳定过程的条件高斯表示以促进仿真，以及用于可扩展优化的对称感知蒙特卡洛估计器。实验表明，在合成数据集与真实数据集上，当基于高斯假设的变分方法失效时，本方法能准确捕捉跳跃动力学并获得可靠的变分后验推断。