AutoSR: Automatic Symbolic Regression by Searching Research States

Scientific discovery from data requires finding a mathematical relation that accurately accounts for a phenomenon and remains valid beyond the immediate observed sample. Traditional symbolic regression methods typically search for explicit mathematical expressions and rank them solely by numerical fit and syntactic complexity. However, finite and noisy data can yield multiple structurally distinct expressions that achieve nearly identical error levels while implying vastly different physical behaviors outside the observed regime.

Scientists address this underdetermination by applying domain knowledge, causal expectations, and physical priors to judge plausibility, test hypotheses, and guide subsequent investigations. Existing machine-learning and automated approaches often lose this crucial scientific record—such as motivations, failed alternatives, and diagnostics—once an equation is generated. To bridge this gap, the authors propose Automatic Symbolic Regression (AutoSR), a fully automated system that performs symbolic regression by searching over persistent scientific investigations.

AutoSR extends symbolic regression from equation-level search to automated scientific investigation through two primary innovations: Research States and Research-State Search. Rather than treating candidate equations in isolation, AutoSR packages each research attempt into a Research State. This persistent object couples a candidate equation with its scientific motivation, computational evidence, execution artifacts, and an independent review record.

The system organizes these Research States into a branching search tree using progressive-widening Monte Carlo tree search (PW-MCTS). Proposer and reviewer agents interact along each branch to generate hypotheses, test them computationally, and critique the results. Child states inherit the accumulated research history of their ancestors along a specific path, allowing promising lines of inquiry to build upon past findings while sibling branches independently explore alternative explanations. Ultimately, the accumulated record is synthesized into an auditable final report that explains the leading relation and the basis for its selection.