FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations
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.
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BibTeX
@misc{liu2026foundpde,
title = {{FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations}},
author = {Langchen Liu and Junlin Luo and Min Zhu and Zhuoran Yang and Lu Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26963},
eprint = {2610.26963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26963},
}
RIS
TY - UNPB
TI - FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations
AU - Langchen Liu
AU - Junlin Luo
AU - Min Zhu
AU - Zhuoran Yang
AU - Lu Lu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.26963
UR - https://arxiv.org/abs/2610.26963
AB - 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.
ER -