Source-linked AI summary
AutoSR: Automatic Symbolic Regression by Searching Research States
Kejia Zhang, Youran Sun, Xinyu Ren, Chugang Yi, Haizhao Yang
TL;DR
Finite noisy data can support competing equations, while conventional symbolic regression often preserves only formulas and scores. AutoSR searches persistent research investigations that retain reasoning, evidence, and review, recovering algebraically equivalent relations on all nine selected benchmark challenges.
Problem
Finite noisy data can support structurally different equations with similar fit but divergent scientific implications, while conventional symbolic regression retains limited research context.
Method
AutoSR couples candidate equations with persistent reasoning, computational evidence, failed tests, and review, then allocates investigation through progressive-widening MCTS.
Results
AutoSR recovered algebraically equivalent relations on all nine selected challenge problems across two benchmark suites, including three cp3-bench equations unrecovered by twelve published systems.
Takeaways & Limitations
Persistent research records let evidence influence subsequent investigations and connect the leading relation to its reasoning, limitations, history, and alternatives.
Takeaways & Limitations
Evidence is limited to nine selected challenge cases with one recovered trajectory per case, so benchmark-wide accuracy and repeated-run reliability remain unestablished.
Abstract
from arXiv · showhide
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 \textbf{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.
1 Introduction
AutoSR reframes symbolic regression as automated scientific investigation: persistent Research States preserve reasoning, evidence, failed attempts, and review while scientific priors and accumulated evidence guide search. Across nine selected benchmark challenges, it recovers algebraically equivalent ground-truth relations in every case, including three equations unrecovered by twelve published systems.
- Research-State Search: AutoSR represents each candidate investigation as a persistent Research State containing its equation and broader scientific record.The record includes motivations, experiments, computational evidence, failed alternatives, diagnostics, and reviewer criticism.
- Scientific investigation: Scientific priors guide hypothesis proposal, computational testing, criticism, and acceptance rather than only evaluating candidate equations.This operationalizes the scientific inquiry cycle, in which hypotheses are tested against evidence and requirements beyond numerical fit.
- Global search: Independent branches let alternative explanations develop separately, while accumulated findings determine subsequent investigations instead of numerical fit alone.Descendant states build on prior evidence, and separate branches avoid prematurely reducing alternatives to one expression.
- System design: AutoSR autonomously generates hypotheses, runs computational tests, conducts independent review, and selects subsequent investigations under a supplied problem specification and computation budget.The final report connects the leading equation to supporting evidence, limitations, research history, and credible alternatives.
- Evaluation: 9 selected benchmark problems yielded algebraically equivalent ground-truth relations in every case, including 3 cp3-bench equations unrecovered by any of 12 published SR systems.The evaluation also covered six structurally diverse LSR-Transform equations using the same core search procedure and agent roles.
2 Related Work
Related work spans expression-based symbolic regression, learned proposal mechanisms, formalized scientific priors, Monte Carlo tree search, and increasingly agentic research workflows. AutoSR builds on these directions by using progressive-widening search over persistent Research States that preserve reasoning, evidence, review, and scientific knowledge beyond a single objective.
- Classical symbolic regression: Symbolic regression searches jointly over expression structure and parameters, evolving expression trees through variation and fitness-based selection.Early scientific-discovery systems extended this search with active dynamical-system probing and invariant-relation discovery.
- Model selection: Candidates are typically selected using predictive error alongside parsimony, sparsity, structural priors, or related model-selection criteria.SRBench shows that predictive accuracy, symbolic recovery, and expression complexity capture different performance dimensions.
- Scientific knowledge and constraints: Scientific prior knowledge has been formalized through constraints on data, bounds, derivative shape, dimensional consistency, structural hypotheses, probabilistic priors, and background axioms.AutoSR also admits natural-language knowledge, which can influence hypothesis formation, computational testing, and criticism rather than only a fixed objective.
- Learned and agentic methods: Learned and language-model-based systems increasingly propose, refine, evaluate, and retain equations using neural priors, synthetic pretraining, diagnostics, experience buffers, and structure libraries.AutoSR specializes the broader movement toward research agents by defining Research State contents, inheritance, independent review, and search semantics for equation discovery.
- Tree search: MCTS allocates simulations through selection, expansion, evaluation, and backup; UCT balances exploration and exploitation, while progressive widening handles impractical action spaces.Symbolic Physics Learner applies MCTS to construct mathematical expression trees.
3 Method
AutoSR treats symbolic regression as budgeted scientific investigation: it searches persistent Research States that combine candidate relations with reasoning, evidence, and review. PW-MCTS allocates asynchronous proposer–reviewer expansions, and the final report explains the leading relation alongside alternatives and search history.
- Problem interface: AutoSR takes a problem specification and computation budget, returning both a leading relation and a final report connecting it to evidence, limitations, history, and alternatives.The specification includes data, variable meanings, supporting evidence, evaluation criteria, and requirements for acceptable relations.
- Problem interface: The problem interface combines problem.md, data, and supporting files containing scientific context, explicit requirements, and potentially checkable constraints such as dimensional consistency or asymptotic behavior.Other priors, including causal expectations, ontology, and mechanistic plausibility, may remain difficult to verify directly.
- Research-State search: PW-MCTS selects which investigation to initiate or extend, balancing new branches against promising continuations while allowing multiple long-running proposer–reviewer expansions asynchronously.Progressive widening admits more independent or descendant investigations as completed and pending evidence accumulates.
- Research States: Each Research State records a candidate relation, motivation, computational artifacts, review, score, and parent, preserving the evidence needed to interpret its scalar search score.Artifacts can include code, execution results, fitted parameters, figures, residual analyses, diagnostics, and unsuccessful tests.
- Proposer–reviewer loop: Proposers develop and test hypotheses through code, fitting, diagnostics, and revision, while reviewers rerun computations and check criteria, residuals, leakage, hidden parameters, requirements, and unsupported conclusions.Reviewer-only held-out data and hidden instructions can prevent proposers from optimizing directly against them.
- Final report: The final report explains the leading result and reconstructs the search landscape by grouping expression families, tracing evolution paths, and presenting the next four candidates as alternatives.It also summarizes the evidence associated with each family and the parent relations connecting states.
4 Experiments
Across nine selected challenges from cp3-bench and LSR-Transform, AutoSR recovered algebraically equivalent relations while using a common search procedure and agent roles. These cases demonstrate capability on difficult structures, not benchmark-wide accuracy, repeated-run reliability, component contributions, or total system cost.
- Experimental procedure: The same core search procedure and role prompts were used for every task while the ground-truth equation remained hidden until the search ended.An external evaluator checked algebraic equivalence after each run, and internet access was prohibited.
- Overall evaluation: AutoSR recovered algebraically equivalent relations on all nine selected challenges spanning two benchmark suites and diverse structural difficulties.The evaluation reports one recovered trajectory per task, so the results demonstrate capability on selected cases rather than benchmark-wide accuracy or stochastic reliability.
- cp3-bench: 3 of 3 selected cp3-bench problems were recovered, whereas all twelve published methods recovered zero corresponding relations.The selected tasks involve dependent input features, unified cusp/core density profiles, and a strongly oscillatory gravitational-wave signal with an additional mass parameter.
- LSR-Transform: 6 of 6 selected LSR-Transform tasks were recovered despite requiring substantially different transformations.The tasks cover nested radicals, trigonometric inversion, fractional powers, paired exponentials, high-dimensional rational expressions, and logarithmic inversion.
- Interpretation and limitations: The nine cases do not estimate benchmark-wide recovery rate, repeated-run success probability, individual component contributions, or total system cost.Reported proposer times exclude other explored branches, reviewer calls, parallel work, conventional-SR runs, and other system costs.
5 Discussion
AutoSR’s discussion emphasizes that its scientific significance comes from searching persistent Research States rather than isolated expressions. These states preserve evidence and criticism, support competing investigations, and connect the selected relation to its research history, limitations, and alternatives in a final report.
- Workflow: AutoSR’s common sustained research workflow and agent roles recover structurally diverse equations across the selected problems.The same core search procedure was retained across these cases.
- Research States: Research-State search preserves motivations, computations, failed tests, and criticism so evidence can guide subsequent investigation and branch-specific memory can develop competing explanations independently.This distinguishes AutoSR from expression-level search, which retains only a formula and its score.
- Final report: Persistent Research States let AutoSR’s final report connect the leading relation to its evidence, limitations, research history, and credible alternatives.AutoSR can continue an investigation automatically after receiving a problem specification and computation budget, while scientists still formulate problems and interpret results.
6 Limitations and Future Evaluation
AutoSR’s current evidence is limited to nine selected cases and does not establish benchmark-wide reliability, computational requirements, or scientific credibility under realistic uncertainty. Future evaluation should use matched ablations, repeated and broader tests, and direct scheduler comparisons.
- Evaluation scope: Nine selected challenge cases with one recovered trajectory each do not establish benchmark-wide accuracy, repeated-run reliability, or computation for specified-probability recovery.Reported proposer times also exclude reviews, unsuccessful and parallel branches, conventional-SR runs, token usage, and orchestration overhead.
- Ablation studies: Matched-budget ablations should separate direct LLM generation, tool use, proposer–reviewer iteration, flat best-of-N Research States, tree search, conventional SR, and supplied scientific context.The proposed comparisons include data and variable names alone versus the full scientific background and requirements.
- Scientific validity: Exact benchmark recovery does not establish discrimination between scientifically credible and numerically competitive relations under noise, extrapolation, incomplete theory, or conflicting evidence.Future tests should assess final reports, alternative relations, preserved search landscapes, newly constructed problems, and private domain data because memorized scientific knowledge cannot be ruled out on established benchmarks.
- Search scheduling: Scheduler evaluation should compare raw reviewer scores with empirical-percentile rewards and sequential UCT with WU-UCT at fixed computation.Measurements should include result quality, redundant concurrent branch selection, worker utilization, and time to the best candidate; ongoing studies will test scientific usefulness across three domains.