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STRIDE: A Self-Reflective Agent Framework for Reliable Automatic Equation Discovery

🕒 Published (v1): 2026-05-18 03:14 UTC · Source: Arxiv · link

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STRIDE: self-reflective agent avoids generation-only loops in symbolic equation discovery

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TL;DR

STRIDE is a multi-role agentic framework for LLM-based symbolic regression that replaces the standard single-loop generation–score–memorize paradigm with four coordinated roles: data-aware generation, mixed-fitting evaluation, critic–executor reflective repair, and semantic memory. By grounding each role in fitted parameter feedback rather than raw generated form, STRIDE recovers more structurally accurate equations and generalizes better to out-of-distribution (OOD) regimes. Ablations confirm that each module contributes independently, with mixed fitting producing the largest single-component gain.

Problem

Standard LLM-based equation discovery (e.g., LLM-SR) collapses generation, evaluation, repair, and memory management into a single score-conditioned loop. This causes three failure modes: (1) promising skeletons are discarded when early parameter fitting underestimates their potential; (2) near-correct equations are abandoned instead of locally repaired; (3) experience libraries accumulate redundant, score-dominated hypotheses that provide little structural diversity to guide future sampling.

Method

STRIDE structures discovery as a closed-loop workflow with four specialized components:

Generator (data-aware sampling): The generator prompt is augmented with lightweight data hints extracted from training data—mean statistics, bias tendency, parity patterns, dominant polynomial/interaction/nonlinear terms—plus elite cases retrieved from semantic memory islands.

Evaluator (mixed fitting): Each skeleton is parsed into an AST and its parameters split into linear coefficients \(w\) and nonlinear parameters \(q\). The evaluator performs nested optimization: an outer loop searches \(q\), and for each \(q\) the inner loop solves \(w^\star(q) = \arg\min_w \|f(u;w,q)-y\|_2^2 + \lambda\|w\|_2^2\), yielding normalized error \(J(q) = \frac{1}{N\,\mathrm{Var}(y)}\|f(u;w^\star(q),q)-y\|_2^2\). BFGS is run as a fallback. The composite score is \(\mathrm{Score}(f,\theta^\star;D) = W_\mathrm{fit}\bigl[-\log(\mathrm{NMSE}+\epsilon)\bigr] - W_\mathrm{comp}\,C(f,\theta^\star)\), penalizing parameter count, sensitivity, and curvature.

Critic–executor (reflective repair): A trigger policy \(\mathrm{TriggerCritic}(s)\) gates reflection on candidates that are promising but imperfect (positive score, below threshold), using a stochastic trigger probability \(\pi_c\). A critic agent diagnoses the failure from fitted parameters and score feedback, proposing actions from \(\mathcal{A} = \{\textsc{Remove}, \textsc{Simplify}, \textsc{Add}\}\); an executor converts these into revised executable candidates. Refined candidates pass a lightweight screening pass before full re-evaluation.

Semantic memory (diversity-preserving updating): Equations are canonicalized, TF-IDF-encoded, and clustered by cosine similarity. Each cluster retains only the highest-scoring elite; structurally similar hypotheses compete within a cluster while diverse structures occupy separate clusters, keeping retrieved in-context exemplars both compact and structurally varied.

Key Contributions

  • Identifies LLM-based equation discovery as a multi-role self-reflective agent problem and formalizes STRIDE as a coordinated four-role closed-loop workflow.
  • Data-aware generator that conditions on extracted statistical properties of training data (parity, dominant terms, bias) to bias structural proposals before fitting.
  • Mixed nested fitting (linear inner loop + nonlinear outer loop + BFGS fallback) that gives the critic–executor grounded parameter-level feedback rather than raw skeleton scores.
  • Critic–executor repair module that converts score-and-parameter feedback into constrained symbolic edits, enabling targeted local revision of near-correct equations.
  • Semantic memory with TF-IDF + cosine-similarity clustering that preserves structural diversity among elite hypotheses, preventing redundancy collapse in the experience buffer.
  • Empirical validation on four LLM-SR benchmarks and four LSR-Synth suites under both ID and OOD settings, with ablations per component and cost analysis.

Results

  • vs. LLM-SR (GPT-5.1): STRIDE improves average \(\mathrm{Acc}_{0.1}\) by +3.45% on four main benchmarks and +5.16% on LSR-Synth; strict \(\mathrm{Acc}_{\max,0.1}\) is best or tied-best across nearly all LSR-Synth domains.
  • vs. LLM-SR (Claude-3-Haiku): STRIDE improves average \(\mathrm{Acc}_{0.1}\) by +2.95% (main) and +11.32% (LSR-Synth).
  • Oscillator 2 OOD (GPT-5.1): STRIDE reaches \(\mathrm{NMSE} = 4.48 \times 10^{-21}\), \(\mathrm{Acc}_{0.001} = 100\%\) vs. LLM-SR at \(3.05\times10^{-3}\), \(19.33\%\).
  • CRK suite (GPT-5.1): \(\mathrm{Acc}_{\max,0.1} = 100\%\), \(\mathrm{NMSE} = 1.83\times10^{-9}\) vs. LLM-SR at \(60\%\), \(7.06\times10^{-5}\).
  • Ablation (Oscillator 2 OOD): Removing mixed fitting degrades \(\mathrm{Acc}_{0.001}\) from \(100\%\) to \(58.67\%\) and raises NMSE to \(1.85\times10^{-4}\); removing critic–executor degrades to \(94.17\%\); removing semantic memory to \(100\%\) (but NMSE rises to \(4.61\times10^{-9}\)).
  • Cost (Table 3): STRIDE uses ~3.5M tokens and ~6.7h per problem, achieving \(3.26\times\) relative \(\mathrm{Acc}_{0.1}\) (best) and \(1\times\) relative NMSE (lowest, i.e., best) vs. comparably budgeted baselines.

Limitations

  • Token and runtime budget (~3.5M tokens, ~6.7h per problem) is comparable to the most expensive baselines (DrSR: ~3.8M, ~7.2h), which may be prohibitive for large-scale or interactive use.
  • Trigger probability \(\pi_c\) and score thresholds for reflection gating introduce hyperparameters whose sensitivity is not fully characterized outside the ablation tasks.
  • TF-IDF-based semantic clustering is a bag-of-symbols approximation; structurally distinct equations sharing operator tokens may be incorrectly co-clustered, or genuinely related equations may be separated.
  • Evaluation is limited to scalar-output symbolic regression; applicability to vector/PDE discovery or noisy, high-dimensional data is not demonstrated.
  • The critic–executor performs constrained local edits (\(\textsc{Remove}/\textsc{Simplify}/\textsc{Add}\)) and may not escape large structural errors requiring global restructuring.

Relevance to Harnesses / Meta-Harnesses

STRIDE is a direct instantiation of a multi-role agentic harness: it decomposes a complex iterative search task (symbolic regression) into specialized sub-agents (generator, evaluator, critic, executor, memory manager) coordinated by explicit feedback channels and trigger policies—exactly the architecture pattern studied in harness and meta-harness research. The critic–executor repair loop, gated by a probabilistic trigger policy, exemplifies a harness-level control flow that decides when to invoke a sub-agent rather than pass through, which is a key design question in meta-harness orchestration. The semantic memory module, acting as a diversity-aware experience store that feeds back into the generator's prompt, demonstrates how shared persistent state across agent roles can improve global search quality—a mechanism directly relevant to long-horizon harness design. For researchers tracking this topic, STRIDE provides a worked example of grounding multi-agent coordination in verifiable, data-derived feedback (fitted scores and parameters) rather than purely model-generated critique, which is a principled approach to reliability in agentic pipelines.