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Discovering multiscale deep formulas in complex systems via neural-guided lambda calculus
Hanqiao Yu1,2, Shusen Yang3,4, Xuebin Ren5,6
1National Engineering Laboratory for Big Data Analytics, Xi'an Jiaotong University, Xi'an, Shaanxi, China.
Nature Communications
|June 16, 2026
Summary
Deflex, a new AI method, automatically extracts multiscale formulas from complex systems. This approach enhances scientific discovery by efficiently identifying scale-specific patterns in diverse systems.
Area of Science:
- Complex Systems Science
- Artificial Intelligence
- Mathematical Modeling
Background:
- Identifying concise mathematical formulas for complex systems is a fundamental scientific challenge.
- Current AI methods struggle with scale-specific formula extraction in multiscale systems.
Purpose of the Study:
- To present Deflex, an end-to-end AI method for automated multiscale formula discovery.
- To extract diverse formulas, including invariants and distributions, from complex systems.
Main Methods:
- Deflex utilizes two subsystems: Deflexformer (a deep energy model) and Deflexpressor (a symbolic regression model).
- Deflexpressor pre-trains Deflexformer using synthetic data, enabling the decoupling of multiscale latent relationships for formula discovery.
Main Results:
- Deflex demonstrates up to 7-fold higher efficiency compared to state-of-the-art methods across six diverse complex systems.
- The method successfully automates the discovery of multiscale formulas, including invariants and distributions.
Conclusions:
- Deflex offers an efficient and automated solution for extracting multiscale formulas from complex systems.
- This AI-driven approach has broad applicability for scientific discovery across various disciplines.
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