Discovering Symbolic Partial Differential Equation by Abductive Learning

En-Hao Gao, Cunjing Ge, Yuan Jiang, Zhi-Hua Zhou

Keywords: Abductive Learning, AI4Science

TL;DR: We propose a dual-driven (data and knowledge) approach for symbolic PDE discovery that exhibits superior performance in terms of noise robustness and hyperparameter stability.

Abstract:

Discovering symbolic Partial Differential Equation (PDE) from data is one of the most promising directions of modern scientific discovery.
Effectively constructing an expressive yet concise hypothesis space and accurately evaluating expression values, however, remain challenging due to the exponential explosion with the spatial dimension and the noise in the measurements.
To address these challenges, we propose the ABL-PDE approach that employs the Abductive Learning (ABL) framework to discover symbolic PDEs.
By introducing a First-Order Logic (FOL) knowledge base, ABL-PDE can represent various PDEs, significantly constraining the hypothesis space without sacrificing expressive power, while also facilitating the incorporation of problem-specific knowledge.
The proposed consistency optimization process establishes a synergistic interaction between the knowledge base and the neural network learning module, achieving robust structure identification, accurate coefficient estimation, and enhanced stability against hyperparameter variation.
Experimental results on three benchmarks across different noise levels demonstrate the effectiveness of our approach in PDE discovery.

Primary Area: Machine learning for sciences (e.g. climate, health, life sciences, physics, social sciences)

Submission Number: 16069