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<p>Symbolic regression, a key problem in discovering physics formulas from observational data, faces persistent challenges in scalability and interpretability. We introduce PhyE2E, an AI framework designed to discover physically meaningful symbolic expressions. PhyE2E decomposes the symbolic regression problem into subproblems via second-order neural network derivatives, and employs a transformer architecture to translate data into symbolic formulas in an end-to-end manner. The generated expressions are further refined via Monte Carlo tree search and genetic programming. We leverage a large language model to synthesize extensive expressions resembling real physics, and train the model to recover these formulas directly from data. Comprehensive evaluations demonstrate that PhyE2E outperforms existing state-of-the-art approaches, delivering superior symbolic accuracy, fitting precision and unit consistency. We deployed PhyE2E on five critical applications in space physics. The AI-derived formulas exhibit excellent agreement with empirical data from satellites and astronomical telescopes. We improved NASA’s 1993 formula for solar activity and provided an explicit symbolic explanation of the long-term solar cycle. We also found that the decay of near-Earth plasma pressure is proportional to the square of the distance <i>r</i> from the Earth’s centre, with subsequent mathematical derivations validated by independent satellite observations. Furthermore, we found symbolic formulas relating solar extreme ultraviolet emission lines to temperature, electron density and magnetic-field variations. The formulas obtained are consistent with properties previously hypothesized by physicists.</p>

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A neural symbolic model for space physics

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Jie Ying,
  • Haowei Lin,
  • Chao Yue,
  • Yajie Chen,
  • Chao Xiao,
  • Quanqi Shi,
  • Yitao Liang,
  • Shing-Tung Yau,
  • Yuan Zhou,
  • Jianzhu Ma

摘要

Symbolic regression, a key problem in discovering physics formulas from observational data, faces persistent challenges in scalability and interpretability. We introduce PhyE2E, an AI framework designed to discover physically meaningful symbolic expressions. PhyE2E decomposes the symbolic regression problem into subproblems via second-order neural network derivatives, and employs a transformer architecture to translate data into symbolic formulas in an end-to-end manner. The generated expressions are further refined via Monte Carlo tree search and genetic programming. We leverage a large language model to synthesize extensive expressions resembling real physics, and train the model to recover these formulas directly from data. Comprehensive evaluations demonstrate that PhyE2E outperforms existing state-of-the-art approaches, delivering superior symbolic accuracy, fitting precision and unit consistency. We deployed PhyE2E on five critical applications in space physics. The AI-derived formulas exhibit excellent agreement with empirical data from satellites and astronomical telescopes. We improved NASA’s 1993 formula for solar activity and provided an explicit symbolic explanation of the long-term solar cycle. We also found that the decay of near-Earth plasma pressure is proportional to the square of the distance r from the Earth’s centre, with subsequent mathematical derivations validated by independent satellite observations. Furthermore, we found symbolic formulas relating solar extreme ultraviolet emission lines to temperature, electron density and magnetic-field variations. The formulas obtained are consistent with properties previously hypothesized by physicists.