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

Abstract: LLM-based equation discovery offers a promising route to recovering symbolic laws from data, but many systems still rely on generation-centered loops that propose candidates, fit parameters, score results, and reuse selected examples. Such loops can misjudge useful skeletons under unreliable fitting, discard near-correct equations that require repair, and accumulate redundant memories that provide limited guidance. We propose STRIDE, a self-reflective agent framework that improves reliability by coordinating data-aware generation, mixed-fitting evaluation, critic--executor repair, and diversity-preserving semantic memory. By turning fitted scores and candidate behavior into shared feedback, STRIDE enables equations to be proposed, assessed, refined, and reused within a closed-loop discovery process. Experiments on representative symbolic-regression benchmarks and LSR-Synth suites show that STRIDE improves accuracy, OOD robustness, and structural recovery across multiple LLM backbones, with ablations and analyses confirming the contribution of its core components.

• The paper introduces STRIDE, a self-reflective, multi-agent framework that decomposes equation discovery into coordinated roles for higher accuracy and reliability.
• It employs data-aware generation, mixed parameter fitting, reflective repair, and semantic memory to iteratively refine and recover structural equations.
• Empirical results show that STRIDE outperforms baselines in robustness and precision, even under challenging out-of-domain conditions.

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

Overview

The paper "STRIDE: A Self-Reflective Agent Framework for Reliable Automatic Equation Discovery" (2605.17790) proposes an agentic architecture for symbolic regression using LLMs, with the principal aim of robust and interpretable equation discovery from empirical data. The central motivation is to address inherent limitations in generation-centered iterative loops that dominate previous LLM-based equation discovery systems—chiefly, unreliability in parameter fitting, the discarding of near-miss equations, and redundancy in memory utilization. STRIDE introduces a multi-agent framework that decomposes the search and refinement process into distinct but tightly coordinated roles: data-aware generation, mixed-parameter fitting, critic-driven local repair, and diversity-preserving semantic memory. This closed-loop, self-reflective approach is positioned as a superior alternative for achieving both higher accuracy and structural robustness, especially under distributional shifts.

Methodological Innovations

Multi-Role Agentic Workflow

STRIDE operationalizes equation discovery as a workflow of four explicit, interacting agents:

• Generator (Data-Aware Proposal): Utilizes lightweight data analytics to extract inductive biases (e.g., mean, bias, parity, dominant terms) and hybrids these with task and exemplar-driven context for prompt construction, guiding the LLM to propose skeleton equations structurally aligned with observed phenomena.
• Evaluator (Mixed Parameter Fitting): Segregates parameters into linear and nonlinear via AST analysis, enabling an optimized two-level fitting regime. Linear coefficients are resolved via ridge least squares conditioned on candidate nonlinear settings, while nonlinear parameters are searched using Powell’s method with BFGS as a fallback. The scoring function penalizes both normalized mean-squared error (NMSE) and complexity (measured via active parameters, sensitivity, and curvature), producing richer feedback for downstream reflection.
• Critic-Executor (Reflective Repair): Triggered by probabilistic gates on promising candidates, the critic inspects fit results and proposes local symbolic edit actions (REMOVE, SIMPLIFY, ADD). The executor operationalizes these edits into revised equations, which undergo rapid screening before comprehensive re-evaluation. Only the highest-scoring candidate following this loop is retained.
• Semantic Memory (Diversity-Preserving Storage): Employs TF-IDF vectorizations and cosine-similarity–based clustering to prevent redundant storage, maintaining a portfolio of structurally diverse, high-scoring equations. Memory retrieval for prompt construction focuses on representative cluster elites, enhancing coverage and sample efficiency.

Closed-Loop Feedback and Iterative Refinement

The core technical advance is the integration of parameter-fitting feedback as a cross-agent signal: generation, evaluation, critique, and revision are tightly coordinated, with fitted behavioral evidence informing every stage. This contrasts with prior approaches that rely exclusively on symbolic form or shallow error-based selection, which are prone to discarding correct but poorly fit structures.

Empirical Results

Benchmark Performance

STRIDE is evaluated on a suite of established symbolic regression and equation discovery benchmarks (Oscillator, E. coli Growth, Stress-Strain, CRK, PO, MatSci, BPG), using both in-domain (ID) and out-of-domain (OOD) partitions. Multiple LLM backbones (GPT-5.1, Claude-3-Haiku) are tested for generality.

• Accuracy: Across all tested domains, STRIDE delivers either best or tied-best results for NMSE and tolerance-based accuracy ([email protected], [email protected]), frequently achieving exact recovery of ground truth under strict criteria. Notably, on the hardest LSR-SYNTH domains CRK and BPG, STRIDE sustains high OOD [email protected], and in many cases, NMSE drops below $10^{-9}$ even out-of-domain.
• Robustness: Unlike baselines, which often degrade under distributional shift, STRIDE’s equation forms preserve extrapolative fidelity, indicating true structural recovery beyond mere in-sample fit.
• Component Ablation: Removal of any core module—data hints, mixed fitting, critic-executor, or semantic memory—degrades reliability, with the absence of mixed fitting yielding the strongest collapse in OOD accuracy, confirming the critical role of informed parameter separation and feedback-driven repair.

Reflection and Memory Diversity

The critic-executor reflection loop, activated conditionally, is shown to convert parameter and structural feedback into significant post-hoc improvements on promising candidates. Semantic memory analysis reveals increased retention of singleton or small structurally diverse high-score clusters, directly addressing catastrophic forgetting and replication risks present in score-only memory policies.

Cost and Efficiency

STRIDE does not entail prohibitive computational or token costs compared to other leading baselines; the net inference budget is effectively amortized by the higher success rate and search efficiency facilitated by the reflective agentic architecture.

Theoretical and Practical Implications

The transition from generation-centered search to a decomposed, feedback-coupled multi-agent architecture establishes a new operational paradigm for LLM-augmented scientific discovery. STRIDE’s design draws from agentic RL literature, but its implementation demonstrates that robust equation discovery hinges not only on candidate diversity or search heuristics but fundamentally on integrating parameter feedback into all stages of the symbolic search. This design principle generalizes as follows:

• Robustness to Data Regimes: Explicitly using fitted behavior (not just structural priors) ensures that near-miss candidates are salvageable, mitigating the risk of overfitting and selection bias especially in low-sample and high-noise settings.
• Reusability and Interpretability: Semantic memory’s organization of knowledge into structurally diverse, high-score exemplars directly translates to improved knowledge transfer and interpretability, both necessary for real-world scientific adoption.
• Scalability: The modular agentic workflow, with critic/executor loops and memory abstraction, is naturally extensible—future developments may incorporate more advanced data-hint extraction, active learning strategies for parameter space exploration, or causal/physical prior integration.

Limitations and Future Directions

STRIDE’s current limits reflect open research frontiers:

• Parameter Role Identification: Heuristic AST-based approaches may misclassify parameter types in deep or tangled expressions, suggesting the need for automated symbolic analysis tools or neural-guided type inference.
• Semantic Equivalence: Clustering based on surface-form similarity (TF-IDF) fails to perfectly align structure equivalence classes; embedding techniques or formal symbolic canonicalization may address these issues.
• Real-World Noisy Data: While tested extensively on clean, ground-truth–available data, practical deployments will require adaptive uncertainty quantification and domain-aware validation mechanisms.

Conclusion

STRIDE (2605.17790) establishes a rigorous, modular agentic framework for scientific equation discovery with LLMs. Through coordinated data-aware prompting, mixed parameter fitting, reflective repair, and diversity-preserving semantic memory, it demonstrably advances accuracy, OOD robustness, and structural recovery across symbolic regression tasks. The architecture offers a template for future research into interpretable, autonomous scientific discovery systems, with substantial space for subsequent augmentations in parameter analysis, memory abstraction, and feedback-driven symbolic reasoning.