Ordinary differential equations (ODEs) underlie dynamical systems which serve as models for a vast number of natural and social phenomena. Yet inferring the ODE that best describes a set of noisy observations on one such phenomenon can be remarkably challenging, and the models available to achieve it tend to be highly specialized and complex too. In this work we propose a novel supervised learning framework for zero-shot inference of ODEs from noisy data. We first generate large datasets of one-dimensional ODEs, by sampling distributions over the space of initial conditions, and the space of vector fields defining them. We then learn neural maps between noisy observations on the solutions of these equations, and their corresponding initial condition and vector fields. The resulting models, which we call foundational inference models (FIM), can be (i) copied and matched along the time dimension to increase their resolution; and (ii) copied and composed to build inference models of any dimensionality, without the need of any finetuning. We use FIM to model both ground-truth dynamical systems of different dimensionalities and empirical time series data in a zero-shot fashion, and outperform state-of-the-art models which are finetuned to these systems. Our (pretrained) FIMs are available online

翻译：常微分方程是构成动态系统的基础，这些系统广泛用于模拟大量自然和社会现象。然而，从一组带有噪声的观测数据中推断出最能描述某一现象的常微分方程极具挑战性，现有的模型往往高度专业化且复杂。在这项工作中，我们提出了一种新的监督学习框架，用于从噪声数据中进行常微分方程的零样本推理。我们首先通过采样初始条件空间和定义向量场的空间分布，生成大规模的一维常微分方程数据集。然后，我们学习这些方程解的噪声观测与其对应初始条件及向量场之间的神经映射。由此产生的模型，我们称之为基础推理模型，具有以下特点：(i) 可沿时间维度复制和拼接以提高分辨率；(ii) 无需任何微调即可通过复制和组合构建任意维度的推理模型。我们使用FIM以零样本方式对不同维度的真实动态系统和经验时间序列数据进行建模，并超越了针对这些系统进行微调的最先进模型。我们的（预训练）FIM模型已在线公开。