Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System¶

🕒 Published (v1): 2026-07-15 08:56 UTC · Source: Arxiv · link

Ask a follow-up

Open an assistant pre-loaded with this paper's context.

TL;DR¶

MEDA (Mechanistic Equation Discovery Agentic system) is a multi-agent harness that couples LLM agents with constrained symbolic regression (SR) to discover ODE models of biological dynamical systems. It orchestrates a pipeline of specialized agents—Literature Surveyor, Formalizer, Runner, Reporter, Recapper, Evaluator—each reading/writing structured artifacts to a shared session directory. Ablations show that knowledge-guided formalization, not numerical fitting, is the load-bearing component for structural recovery.

Problem¶

Existing equation-discovery approaches either narrow-focus on benchmarked SR search or over-promise with broad end-to-end automation pipelines, while neglecting the full scientific reasoning cycle: variable definition, mechanistic constraint generation, candidate ODE proposal, and interpretability-aware evaluation. Biological systems are especially underserved because their data are noisy, sparse, and heterogeneous, and their prior knowledge is incomplete rather than formally expressible as symmetries or reduction laws (as in physics).

Method¶

MEDA operates as an orchestrator-coordinated set of specialized agents, each with a defined input artifact and a structured output artifact written to a shared session directory:

Literature Surveyor: retrieves cited background, expected constraints, candidate model families, and qualitative dynamical signatures → lit_review.md
Formalizer: converts literature into a machine-readable problem specification with typed hard/soft constraints, seed equations, polynomial-degree limits, and constraint weights → constraints.md, problem.json
Explorer (data-anchored mode only): performs EDA and a SINDy-based sparse-regression screen on the supplied time series to identify data-supported terms
Runner: executes a genetic algorithm over sparse polynomial ODE systems with a composite objective: constraint satisfaction + trajectory fit (when data present) + sparsity; hard-constraint gating and core-term protection preserve biological admissibility → symbolic_regression.md
Reporter / Recapper / Evaluator: translate the selected model to human-readable output, diagnose pipeline-level failure modes, and score against quarantined ground truth using variable F1, constraint F1, term P/R/F1, expression score (\(1 - \text{TED}\)), inclusion score, and expert plausibility score

Two modes exist: constraint-only (literature-derived constraints and parsimony only) and data-anchored (trajectory fit added to search objective). Ground-truth equations are quarantined from all discovery agents and accessed only by the post-hoc Evaluator.

Key Contributions¶

A full agentic harness for biological ODE discovery that reconstructs the scientific reasoning process (literature → formalization → search → critique → evaluation) rather than treating equation discovery as a one-step benchmark
A typed constraint grammar that encodes hard and soft biological admissibility conditions, enabling mechanistic gating during symbolic search
Three-tier evaluation protocol: information retrieval (IR, canonical model reconstruction), extrapolation (EX, reasoning to unseen variants), and open-ended discovery (DI)
Expert plausibility scoring: average of five biomathematicians with PhD-level expertise, providing a domain-grounded quality signal beyond symbolic accuracy
Ablation and robustness analysis isolating the contributions of literature retrieval, EDA/SINDy initialization, formalization, and pure numerical fitting

Results¶

Full system (with data, ablation over IR/EX/DI): average term-level F1 = 1.00, plausibility \(S_{\text{plaus}}\) = 0.967
No-EDA/SINDy variant: term F1 = 1.00, plausibility = 0.969 — nearly identical to full system; EDA is not load-bearing
No-literature variant: term F1 = 0.732, plausibility = 0.700 — removing literature retrieval degrades structural recovery; misinformation dynamics plausibility drops from 0.98 → 0.59
Pure-SINDy variant: term F1 = 0.541, term precision = 0.382, plausibility = 0.603 — largest degradation; introduces many false-positive terms and violates conservation constraints in SIR-type systems
Noise robustness: classical SIR retains term F1 = 1.00 through 30% proportional Gaussian noise, degrading to 0.73 at 40–60% (increased spurious cross-terms, not loss of reference terms); age-structured SIR maintains F1 ≥ 0.90 across all tested noise levels
Data can hurt: FitzHugh–Nagumo adaptation and chronic wound healing show that adding data introduces high-degree spurious terms while missing defining biological couplings; the host–dual-pathogen case shows constraint-generation errors that data fitting cannot correct downstream

Limitations¶

Evaluated only on polynomial ODE libraries; non-polynomial dynamical terms (e.g., Hill functions, sigmoidal kinetics) are outside scope
Expert plausibility scoring is averaged across five reviewers but remains subjective and domain-specific; interrater agreement is not reported
LLM-based stages (literature retrieval, formalization) introduce stochastic variation not fully characterized; prompt-perturbation robustness is acknowledged but deferred to separate analysis
When constraint generation produces a wrong biological family (host–dual-pathogen), downstream numerical fitting cannot recover the correct structure—error propagation through the pipeline is not mitigated
No comparison against the strongest published LLM-SR hybrid baselines (e.g., LLM-SR) on shared benchmarks, making direct state-of-the-art positioning difficult

Relevance to Harnesses / Meta-Harnesses¶

MEDA is a concrete harness architecture: specialized agents are composed in a defined sequence, communicate exclusively through structured file artifacts in a shared session directory, and are coordinated by an orchestrator—a design pattern directly analogous to agent harnesses in software engineering contexts. The paper provides empirical evidence for a key harness-design principle: the scaffolding layer (knowledge-guided formalization, constraint grammar, artifact validation) matters more than the raw compute in individual stages (numerical fitting, SINDy initialization), which informs how to allocate design effort in multi-agent harnesses. The quarantine of ground-truth data until post-hoc evaluation is a reproducibility and integrity pattern applicable to any harness that produces verifiable outputs. The ablation methodology—systematically dropping pipeline stages to measure their marginal contribution—offers a reusable evaluation template for harness component attribution.