FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations

Langchen Liu, Junlin Luo, Min Zhu, Zhuoran Yang, Lu Lu

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…