Symbolic regression discovers accurate and interpretable formulas to describe given data, thereby providing scientific insights for domain experts and promoting scientific discovery. However, existing symbolic regression methods often use complexity metrics as a proxy for interoperability, which only considers the size of the formula but ignores its internal mathematical structure. Therefore, while they can discover formulas with compact forms, the discovered formulas often have structures that are difficult to analyze or interpret mathematically. In this work, inspired by the observation that physical formulas are typically numerically stable under limited calculation precision, we propose the Effective Information Criterion (EIC). It treats formulas as information processing systems with specific internal structures and identifies the unreasonable structure in them by the loss of significant digits or the amplification of rounding noise as data flows through the system. We find that this criterion reveals the gap between the structural rationality of models discovered by existing symbolic regression algorithms and real-world physical formulas. Combining EIC with various search-based symbolic regression algorithms improves their performance on the Pareto frontier and reduces the irrational structure in the results. Combining EIC with generative-based algorithms reduces the number of samples required for pre-training, improving sample efficiency by 2~4 times. Finally, for different formulas with similar accuracy and complexity, EIC shows a 70.2% agreement with 108 human experts' preferences for formula interpretability, demonstrating that EIC, by measuring the unreasonable structures in formulas, actually reflects the formula's interpretability.

In this study, we unveil a new AI model, termed PhyE2E, to discover physical formulas through symbolic regression. PhyE2E simplifies symbolic regression by decomposing it into sub-problems using the second-order derivatives of an oracle neural network, and employs a transformer model to translate data into symbolic formulas in an end-to-end manner. The resulting formulas are refined through Monte-Carlo Tree Search and Genetic Programming. We leverage a large language model to synthesize extensive symbolic expressions resembling real physics, and train the model to recover these formulas directly from data. A comprehensive evaluation reveals that PhyE2E outperforms existing state-of-the-art approaches, delivering superior symbolic accuracy, precision in data fitting, and consistency in physical units. We deployed PhyE2E to five applications in space physics, including the prediction of sunspot numbers, solar rotational angular velocity, emission line contribution functions, near-Earth plasma pressure, and lunar-tide plasma signals. The physical formulas generated by AI demonstrate a high degree of accuracy in fitting the experimental data from satellites and astronomical telescopes. We have successfully upgraded the formula proposed by NASA in 1993 regarding solar activity, and for the first time, provided the explanations for the long cycle of solar activity in an explicit form. We also found that the decay of near-Earth plasma pressure is proportional to r^2 to Earth, where subsequent mathematical derivations are consistent with satellite data from another independent study. Moreover, we found physical formulas that can describe the relationships between emission lines in the extreme ultraviolet spectrum of the Sun, temperatures, electron densities, and magnetic fields. The formula obtained is consistent with the properties that physicists had previously hypothesized it should possess.

Artificial intelligence (AI) technology has demonstrated remarkable potential in drug dis-covery, where pharmacokinetics plays a crucial role in determining the dosage, safety, and efficacy of new drugs. A major challenge for AI-driven drug discovery (AIDD) is the scarcity of high-quality data, which often requires extensive wet-lab work. A typical example of this is pharmacokinetic experiments. In this work, we develop a physical formula enhanced mul-ti-task learning (PEMAL) method that predicts four key parameters of pharmacokinetics simultaneously. By incorporating physical formulas into the multi-task framework, PEMAL facilitates effective knowledge sharing and target alignment among the pharmacokinetic parameters, thereby enhancing the accuracy of prediction. Our experiments reveal that PEMAL significantly lowers the data demand, compared to typical Graph Neural Networks. Moreover, we demonstrate that PEMAL enhances the robustness to noise, an advantage that conventional Neural Networks do not possess. Another advantage of PEMAL is its high flexibility, which can be potentially applied to other multi-task machine learning scenarios. Overall, our work illustrates the benefits and potential of using PEMAL in AIDD and other scenarios with data scarcity and noise.