Discovering governing partial differential equations (PDEs) from observed data is a central inverse problem in computational science. The aim is not only to predict system trajectories but also to extract interpretable symbolic laws that can be analyzed and reused. Existing PDE discovery methods rely on hand‑crafted function libraries, accurate numerical derivatives, or per‑equation symbolic search, making them fragile under sparse observations, noisy data, broad equation families, or unseen symbolic compositions.
We introduce FoundPDE, a pretrained data‑to‑symbol foundation model. A Transformer maps discretized spatio‑temporal trajectories to symbolic equations, while Fourier augmentation and first‑order spatial‑derivative augmentation expose complementary trajectory information. Numerical coefficients are then refined, and the model supports few‑shot fine‑tuning and hierarchical discovery to improve generalization to new physics.
On a clean in‑domain benchmark covering 20 PDE families, FoundPDE achieves 97.50% type recognition. When coefficients shift beyond the training support, recognition remains at 90.53%; with 2% multiplicative Gaussian noise on the input, it is 94.43%, dropping to 52.09% at 10% noise. For unseen composite equations, few‑shot fine‑tuning reaches over 90% exact recovery on four held‑out families, and a hierarchical discovery algorithm recovers two‑term composites without any fine‑tuning.
By amortizing symbolic discovery across equation families, FoundPDE shifts PDE discovery from per‑dataset search toward a reusable pretrained model that can extend to new physics with limited additional data, marking a step toward foundation models that uncover interpretable governing laws directly from scientific data.
Review: FoundPDE demonstrates the promise of large‑scale models in symbolic science, especially their robustness to sparse, noisy data and unseen equation structures, offering a new paradigm for interpretable AI research.