arXiv cs.AI· Langchen Liu, Junlin Luo, Min Zhu, Zhuoran Yang, Lu Lu·· 12 小时前AI 评分34
FoundPDE：面向偏微分方程数据驱动符号发现的生成式基础模型
FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations
AI 导读
FoundPDE 是一个预训练的数据到符号基础模型，用 transformer 将离散时空轨迹映射为符号方程，并结合傅里叶与一阶空间导数增强及数值系数精修。在覆盖 20 个 PDE 家族的域内基准上，其类型识别率达 97.50%；系数偏移下为 90.53%，2% 高斯噪声下为 94.43%，10% 噪声下降至 52.09%。
正文 · 原文
Abstract:Discovering governing partial differential equations (PDEs) from observed data is a central inverse problem in computational science and engineering: the goal is not only to predict trajectories, but to identify interpretable symbolic laws that can be analyzed and reused. Existing PDE discovery methods often depend on hand-designed libraries, accurate numerical derivatives, or per-equation symbolic search, making them fragile under sparse observations, noisy data, broad equation families, and unseen symbolic compositions. Here, we introduce FoundPDE, a pretrained data-to-symbol foundation model for PDE discovery. FoundPDE maps discretized spatio-temporal trajectories to symbolic equations with a transformer, uses Fourier and first-order spatial-derivative augmentation to expose complementary trajectory information, refines numerical coefficients, and supports downstream extension with improved generalization through few-shot fine-tuning and hierarchical discovery. On a clean in-domain benchmark spanning 20 PDE families, our model reaches 97.50% type recognition. Under coefficient shifts beyond the training support, type recognition remains 90.53%; under multiplicative Gaussian noise on the model input, it is 94.43% at a 2% noise level but falls to 52.09% at 10%. For unseen composite equations, few-shot fine-tuning reaches over 90% exact recovery on four held-out families, and we further develop a hierarchical discovery algorithm that recovers two-term composites without fine-tuning FoundPDE. By amortizing symbolic discovery across equation families, FoundPDE shifts PDE discovery from per-dataset search toward a reusable pretrained model that extends to new physics with limited additional data, a step toward foundation models that uncover interpretable governing laws directly from scientific data.
| Comments: | 42 pages, 13 figures, 12 tables |
| Subjects: | Computational Physics (physics.comp-ph); Artificial Intelligence (cs.AI); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.26963 [physics.comp-ph] |
| (or arXiv:2610.26963v1 [physics.comp-ph] for this version) | |
| https://doi.org/10.48550/arXiv.2610.26963 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Langchen Liu [view email]
[v1]
Sun, 18 Oct 2026 15:40:42 UTC (5,060 KB)
来源：arXiv cs.AI · arxiv.org