Stochastic differential equations (SDEs) describe dynamical systems where deterministic flows, governed by a drift function, are superimposed with random fluctuations, dictated by a diffusion function. The accurate estimation (or discovery) of these functions from data is a central problem in machine learning, with wide application across the natural and social sciences. Yet current solutions either rely heavily on prior knowledge of the dynamics or involve intricate training procedures. We introduce FIM-SDE (Foundation Inference Model for SDEs), a pretrained recognition model that delivers accurate in-context (or zero-shot) estimation of the drift and diffusion functions of low-dimensional SDEs, from noisy time series data, and allows rapid finetuning to target datasets. Leveraging concepts from amortized inference and neural operators, we (pre)train FIM-SDE in a supervised fashion to map a large set of noisy, discretely observed SDE paths onto the space of drift and diffusion functions. We demonstrate that FIM-SDE achieves robust in-context function estimation across a wide range of synthetic and real-world processes -- from canonical SDE systems (e.g., double-well dynamics or weakly perturbed Lorenz attractors) to stock price recordings and oil-price and wind-speed fluctuations -- while matching the performance of symbolic, Gaussian process and Neural SDE baselines trained on the target datasets. When finetuned to the target processes, we show that FIM-SDE consistently outperforms all these baselines.

翻译：随机微分方程（SDEs）描述了确定性流动（由漂移函数主导）与随机波动（由扩散函数决定）叠加的动力学系统。从数据中准确估计（或发现）这些函数是机器学习的核心问题，在自然科学和社会科学中具有广泛应用。然而，现有解决方案要么严重依赖对动力学的先验知识，要么涉及复杂的训练过程。本文提出FIM-SDE（用于SDE的基础推理模型），这是一种预训练的识别模型，能够从含噪声的时间序列数据中，对低维SDE的漂移函数和扩散函数进行准确的上下文（或零样本）估计，并可快速微调至目标数据集。利用摊销推理和神经算子的概念，我们以监督方式（预）训练FIM-SDE，将大量含噪声、离散观测的SDE路径映射到漂移函数和扩散函数空间。我们证明，FIM-SDE在广泛的合成和现实世界过程（从经典SDE系统（如双势阱动力学或弱扰动洛伦兹吸引子）到股价记录、油价和风速波动）中实现了稳健的上下文函数估计，其性能与在目标数据集上训练的符号方法、高斯过程和神经SDE基线模型相当。当微调到目标过程时，FIM-SDE始终优于所有这些基线模型。