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

2026-06-05

AI Summary

Existing AI approaches struggle to automatically discover concise mathematical formulas that capture cross-scale, heterogeneous structures—such as invariants and probability distributions—in complex systems. To address this challenge, this work proposes Deflex, a framework that achieves end-to-end multi-scale deep mathematical formula discovery for the first time. Deflex integrates two components: Deflexformer, a decomposable deep energy-based model that learns unified cross-scale representations and disentangles multi-scale latent relationships to guide formula discovery; and Deflexpressor, which employs neural-guided λ-calculus to generate high-order symbolic expressions for synthetic data pretraining. Experiments across six canonical complex systems demonstrate that Deflex improves formula discovery efficiency by up to 7× over current state-of-the-art methods, substantially overcoming the limitations of single-scale symbolic regression.

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.

Problem

• multiscale
• complex systems
• mathematical formulas
• symbolic regression
• scientific discovery

Innovation

• multiscale formula discovery
• neural-guided lambda calculus
• symbolic regression
• deep energy model
• complex systems