O${}_2$PDE: An End-to-end Framework for Solving Inverse PDE Problems from Observations Alone

Solving inverse PDE problems from observations remains a fundamental challenge in applied mathematics and physics-informed machine learning. Classical approaches, such as system identification and symbolic regression, provide interpretable and closed-form representations but are typically restricted to expert-level skills. Recent advances in machine learning have exhibited promise, but the performance of existing solvers relies heavily on abundant labeled data and domain expert priors, thus hindering practical applications in mathematics and physics due to a lack of accuracy and robustness. This work considers the challenging task of reconstructing the PDE system from observations alone without any domain expert priors. We propose an end-to-end framework, named O${}_2$PDE, for tackling this challenge. The workflow of O${}_2$PDE consists of four modules: Data Encoder that converts observations into latent representations, Function Encoder that encodes symbolic candidate functions to queries, Decoder with Cross-Attention for conditioning queries on observation representations, and Dual Head that jointly predicts structure and coefficient identifications. The whole model is supervised by a combination loss relative to multi-label classification and coefficient regression, with a regularizer that fuses temporal chunks and consistency losses for enforcing prediction consistency across chunks. We conduct experiments on a collection of PDE systems to evaluate the effectiveness and robustness of O${}_2$PDE in comparison with six baselines, which contains $1.65$ million PDE systems and $458.11$ GB observable data. Compared to six baselines, our method achieves the best performance, where O${}_2$PDE improves 91.9% F1 and 52.42% EM and reduces 94.8% L2RE rather than the fine-tuned LLMs, with only 5.1% of the LLM's total parameters. Even corrupting 5% and 10% Gaussian noises to observations, O${}_2$PDE improves F1 by 21.7% and 18.1%, while reducing L2RE by 56.4% and 47.1%, respectively, compared to the SOTA baseline. This is a laudable result, not only confirming the reconstruction advances of O${}_2$PDE but also opening the door of developing foundation models for inverse PDE problem solving.