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arXiv · 2606.07426

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

Hanqiao Yu·Shusen Yang·Xuebin Ren·Cong Zhao
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Abstract

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.

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Hanqiao Yu, Shusen Yang, Xuebin Ren, Cong Zhao. 2026-06-05. Discovering Multiscale Deep Formulas in Complex Systems via Neural-Guided Lambda Calculus. https://arxiv.org/abs/2606.07426

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