用预训练模型直接从噪声轨迹推断微分方程，无需复杂调参。
Foundation Inference Models for Ordinary Differential Equations
- 通过神经算子直接预测向量场，单次前向传播完成推断
- 零样本性能媲美甚至超越现有符号化基线，且更稳定
- 适合无机器学习背景的研究者快速应用到科学建模中
常微分方程（ODE）在科学建模中至关重要，但从噪声轨迹中推断其向量场仍具挑战。现有方法如符号回归、高斯过程和神经ODE通常需要复杂的训练流程或依赖特定先验知识。我们提出FIM-ODE，一种预训练的通用推断模型，通过单次前向传播直接从噪声轨迹预测向量场。该模型在低阶多项式向量场的先验分布上进行预训练，并使用神经算子表示目标场。FIM-ODE在多种情形下实现强零样本性能，匹配甚至优于近期的预训练符号基线ODEFormer，且预训练提供良好初始化，支持快速稳定的微调，显著优于现代神经与高斯过程基线，且无需机器学习专业知识。
原文摘要 · Abstract (English)
Ordinary differential equations (ODEs) are central to scientific modelling, but inferring their vector fields from noisy trajectories remains challenging. Current approaches such as symbolic regression, Gaussian process (GP) regression, and Neural ODEs often require complex training pipelines and substantial machine learning expertise, or they depend strongly on system-specific prior knowledge. We propose FIM-ODE, a pretrained Foundation Inference Model that amortises low-dimensional ODE inference by predicting the vector field directly from noisy trajectory data in a single forward pass. We pretrain FIM-ODE on a prior distribution over ODEs with low-degree polynomial vector fields and represent the target field with neural operators. FIM-ODE achieves strong zero-shot performance, matching and often improving upon ODEFormer, a recent pretrained symbolic baseline, across a range of regimes despite using a simpler pretraining prior distribution. Pretraining also provides a strong initialisation for finetuning, enabling fast and stable adaptation that outperforms modern neural and GP baselines without requiring machine learning expertise.
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