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Discovering multiscale deep formulas in complex systems via neural-guided lambda calculus
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DOI：10.1038/s41467-026-74299-z.png)
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Abstract
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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. The authors developed an end to-end AI framework for discovering multiscale mathematical formulas in complex systems, using representation learning and symbolic regression to extract scale-specific laws, improving efficiency and enabling automated scientific discovery beyond single-scale methods.
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