cs.SCAug 17, 2026

AutoSR: Automatic Symbolic Regression by Searching Research States

Authors: Kejia Zhang, Youran Sun, Xinyu Ren, Chugang Yi, Haizhao Yang

Organizations: Department of Mathematics, University of Maryland, College Park, MD, USA · The Chinese University of Hong Kong, Hong Kong SAR, China

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Abstract

We introduce Automatic Symbolic Regression (AutoSR), a fully automated system that instantiates Research-Space Symbolic Regression by searching persistent scientific investigations rather than isolated equations. Finite, noisy data often yield numerically competitive expressions that imply very different behavior outside the observed regime, making numerical fit and syntactic complexity insufficient measures of scientific credibility. Existing approaches largely focus on improving expressions, yet the search typically retains little beyond the resulting formula and score, losing the scientific record, such as motivations and probes, that inform what to try next. AutoSR preserves this record in a Research State, coupling each candidate equation with the reasoning, computational evidence, and independent review developed along its branch. Proposer--reviewer agents develop these states under progressive-widening Monte Carlo tree search (PW-MCTS), which allocates computation across competing investigations, while the accumulated research record is ultimately synthesized into a final report that explains the leading relation and the basis for its selection. Across nine selected challenges from two benchmark suites, AutoSR recovers algebraically equivalent relations in every case, including three cp3-bench problems that no published system recovers and six structurally diverse LSR-Transform problems. Overall, AutoSR extends symbolic regression from equation-level search toward automated scientific investigation, allowing scientific knowledge and accumulated evidence to shape both what is explored and how the resulting equation is justified.

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Figure 1: AutoSR uses a proposer–reviewer loop to build a tree of Research States with progressive-widening MCTS. The problem specification supplies data, scientific context, and explicit requirements to the investigation. A secondary actor–reviewer loop advances conventional symbolic regression. The Prompt Economy annotation uses the AutoSR-specific reuse metric defined in Equation 3 . The final report combines the strongest candidates, their supporting evidence, and the resulting search landscape.
ProblemBest cp3-bench resultAutoSR resultProposer time
C3gQLattice; MSE 4.7×10−64.7\times 10^{-6} ; ground truth not recoveredAlgebraically equivalent; MSE 9.0×10−199.0\times 10^{-19}8 min 58 s
C5fGPG; MSE 3.3×10−53.3\times 10^{-5} ; ground truth not recoveredAlgebraically equivalent; MSE 6.2×10−326.2\times 10^{-32}15 min 44 s
C6cGPG; MSE 1.1×10−11.1\times 10^{-1} ; ground truth not recoveredAlgebraically equivalent; MSE 6.1×10−306.1\times 10^{-30}33 min 1 s
Table 1: Results on the selected cp3-bench problems. MSE values are reported to two significant digits.
ProblemAutoSR resultProposer time
I.32.17_4_3Algebraically equivalent; NRMSE 1.8×10−71.8\times 10^{-7}16 min 13 s
I.29.16_0_0Algebraically equivalent; NRMSE 1.2×10−71.2\times 10^{-7}12 min 32 s
II.6.15a_2_0Algebraically equivalent; NRMSE 1.0×10−71.0\times 10^{-7}7 min 47 s
II.35.18_0_0Algebraically equivalent; NRMSE 3.6×10−73.6\times 10^{-7}6 min 26 s
II.36.38_3_0Algebraically equivalent; NRMSE 8.0×10−88.0\times 10^{-8}9 min 7 s
III.4.33_3_0Algebraically equivalent; NRMSE 7.2×10−87.2\times 10^{-8}8 min 9 s
Table 2: Results on the selected LSR-Transform problems. NRMSE values are reported to two significant digits.

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