[Paper] Discovering Multiscale Deep Formulas in Complex Systems via Neural-Guided Lambda Calculus

Source: arXiv - 2606.07426v1

Overview

A fundamental problem in science is identifying underlying patterns of complex systems in the form of concise mathematical formulas. Current Artificial Intelligence (AI)-based methods have shown strong performance in single-scale systems, yet remain limited in identifying scale-specific formulas in multiscale complex systems. We present Deflex, an end-to-end AI method to automatically extract multiscale formulas with potentially different forms, including invariants and distributions, from complex systems. Deflex consists of two subsystems named Deflexformer and Deflexpressor. Deflexpressor is a lambda-calculus symbolic regression model for higher-order formulas. Deflexformer is a decomposable deep energy model for learning unified representations across scales. Deflexpressor generates synthetic data to pre-train Deflexformer, which then guides formula discovery by decoupling multiscale latent relationships. Across six representative complex systems with diverse behaviors, Deflex achieves up to 7-fold higher efficiency than the state-of-the-art methods while enabling automated multiscale discovery. Our work could be a useful tool for scientific discovery across disciplines.

Key Contributions

This paper presents research in the following areas:

• cs.LG

Methodology

Please refer to the full paper for detailed methodology.

Practical Implications

This research contributes to the advancement of cs.LG.

Authors

• Hanqiao Yu
• Shusen Yang
• Xuebin Ren
• Cong Zhao

Paper Information

• arXiv ID: 2606.07426v1
• Categories: cs.LG
• Published: June 5, 2026
• PDF: Download PDF