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arxiv: 2606.25039 · v1 · pith:RS3YL6XWnew · submitted 2026-06-23 · 💻 cs.LG · cs.AI· cs.CL· math.DS

LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search

Nikhil Abhyankar , Sha Li , Sanchit Kabra , Naren Ramakrishnan , Yulia Gel , Chandan K. Reddy This is my paper

Pith reviewed 2026-06-25 23:33 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CLmath.DS
keywords dynamical systemsODE discoverysymbolic regressionactive learninglarge language modelsclosed-loop optimization
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The pith

LLM-ACES recovers governing ODEs by letting an LLM partition the search space and using model disagreement to select new trajectories in a closed loop.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that static methods for discovering ODEs from fixed data often fail because limited trajectories leave multiple equations indistinguishable. LLM-ACES instead runs an iterative loop: an LLM suggests operator priors that split the hypothesis space into regions, equations are fitted inside each region, and disagreement among the fitted candidates determines which new state-space trajectories to measure next. On 122 benchmark systems the method reports the lowest median normalized mean squared error and recovers the correct symbolic form more often than prior approaches. The same loop also reaches strong performance with only one-tenth the data volume and remains accurate when noise is added.

Core claim

LLM-ACES jointly optimizes symbolic hypothesis construction and adaptive data acquisition: the LLM proposes operator priors that partition the large search space into distinct regions; candidate equations are fit to the observed data inside each region; disagreement among these candidates guides the acquisition of informative new trajectories; the process repeats, refining both the hypothesis space and the recovered dynamics.

What carries the argument

LLM-proposed operator priors that partition the search space, combined with disagreement among fitted candidates to select the next trajectories in a feedback loop.

If this is right

  • The closed loop yields the lowest median NMSE across the 122 systems while reaching 46.2 percent and 52.4 percent symbolic accuracy on the two benchmark collections.
  • Performance stays superior when only one-tenth the usual data volume is supplied.
  • Under added noise the method recovers the true symbolic structure instead of introducing spurious terms that fit locally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same disagreement signal could be used to decide when to stop collecting data rather than running a fixed budget.
  • The operator-prior partitioning step may transfer to discovering other classes of governing equations such as delay differential equations or stochastic differential equations.
  • Because the loop explicitly targets identifiability gaps, it could be paired with experimental design methods that already optimize for parameter uncertainty.

Load-bearing premise

LLM-proposed operator priors meaningfully partition the search space and disagreement among fitted candidates reliably identifies the most informative next trajectories.

What would settle it

On the same 122 ODE systems and identical data budgets, replace the LLM-guided acquisition step with random or fixed sampling and measure whether median NMSE and symbolic accuracy remain within one order of magnitude of the reported LLM-ACES results.

Figures

Figures reproduced from arXiv: 2606.25039 by Chandan K. Reddy, Naren Ramakrishnan, Nikhil Abhyankar, Sanchit Kabra, Sha Li, Yulia Gel.

Figure 1
Figure 1. Figure 1: Overview of LLM-ACES. LLM-ACES couples LLM-guided hypothesis-space construction with active trajectory acquisition. (A) The LLM induces constrained symbolic search spaces from task information and prior feedback. (B) Candidate equations are generated and optimized within these spaces. (C) New initial conditions are selected by maximizing predictive disagreement among candidate rollouts. (D) Acquired trajec… view at source ↗
Figure 2
Figure 2. Figure 2: Ablation study of LLM-ACES on 15 ODE benchmark systems from ODEBench. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Discovered equations for the Schnackenberg (top) and Maxwell-Bloch (bottom) systems. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Error analysis comparing Qwen3-32B on 120 Feynman problems versus 63 ODEBench datasets. Many benchmark systems correspond to canonical equations that are likely to have appeared during pretraining, leading to outputs that reproduce verbatim content [12, 17]. To disentangle direct dataset recall from genuine discovery, we anonymize ODEBench and ODEBase by replacing all state variable names, time derivatives… view at source ↗
Figure 5
Figure 5. Figure 5: Sample efficiency comparison. LLM-ACES achieves the lowest error using only 100 training samples, while competing baselines are provided with progressively larger sample budgets. We evaluate the sample efficiency of LLM-ACES on a set of 15 randomly selected datasets, sampled uniformly across dimensions (five each from 1D, 2D, and 3D systems). We compare LLM-ACES against a diverse set of baselines, includin… view at source ↗
Figure 6
Figure 6. Figure 6: Prompt example of LLM-ACES for 3D ODE discovery. The prompt includes instructions, a task specification describing the objective, system characteristics, and data samples, as well as the target function and required structured output format. It also shows intermediate elements such as the current best equations, existing concepts, and failure cases. The prompt shown in [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 7
Figure 7. Figure 7: Prompt example of LLM-ACES for hypothesis exploration. The prompt specifies the task specification, current best equations, existing operator set, and instructs the model to generate a new equation concept. This prompt enhances the diversity of operators. Data Acquisition via Active Experiment Selection. To address the fundamental limitation of static datasets, LLM-ACES incorporates an active learning mech… view at source ↗
Figure 8
Figure 8. Figure 8: Distribution of reconstruction, generalization, out-of-distribution NMSE, and expression complexity across methods for ODEBench datasets. 10 −30 10 −20 10 −10 10 0 SINDy ODEFormer Operon PySR E2E QBC BO APPS-ODE LLM-ODE LLM-Only LLM-ACES Reconstruction 10 −30 10 −15 10 0 Generalization 10 −30 10 −15 10 0 10 15 Out-of-Distribution 0 50 100 150 Complexity [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of reconstruction, generalization, out-of-distribution NMSE, and expression complexity across methods for ODEBase datasets. D Additional Analyses D.1 Robustness to Noise and Irregular Sampling To assess robustness under realistic observation noise, we follow the MDBench [5] evaluation protocol by corrupting clean trajectories at multiple SNR levels and evaluating the discovered right￾hand side… view at source ↗
Figure 10
Figure 10. Figure 10: Robustness under noisy observations. D.2 Data Acquisition for Dynamical Equation Discovery Most existing equation discovery methods operate under a static-data paradigm, implicitly assuming that the available observations are sufficient for identifying the underlying dynamics. Yet low fitting error alone provides no guarantee of correct structural recovery, as different equations may explain the same obse… view at source ↗
Figure 11
Figure 11. Figure 11: Identifiability gap in passive equation fitting. [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of equation recovery from a single trajectory and multiple trajectories. Trajectory diversity improves performance across methods. Effect of trajectory diversity [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of different trajectory acquisition strategies on 1D, 2D, and 3D ODEBench [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Predictions of LLM-ACES for all equations in ODEBench for the first set of initial conditions. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Predictions of LLM-ACES for all equations in ODEBase for the first set of initial conditions. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
read the original abstract

Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over fixed datasets, assuming that the observed trajectories are sufficiently informative. However, dynamical systems evolve over large state spaces, and limited data can make multiple equations observationally indistinguishable, leading to identifiability gaps and the recovery of incorrect governing equations. To address this, we introduce LLM-ACES, or LLM-guided Active Closed-loop Equation Search, a closed-loop framework that jointly optimizes symbolic hypothesis construction and adaptive data acquisition. In LLM-ACES, a large language model (LLM) proposes operator priors that partition the large search space into distinct regions, within which candidate equations are fit to the observed data. The disagreement among these candidates guides the acquisition of informative trajectories, creating a feedback loop that iteratively refines both the hypothesis space and the discovered dynamics. On 122 ODE systems spanning ODEBench and ODEBase, LLM-ACES achieves the lowest median NMSE, outperforming state-of-the-art baselines by several orders of magnitude while achieving a high symbolic accuracy of 46.2% and 52.4%, respectively. Our analysis further shows that LLM-ACES is sample-efficient, achieving better performance with one-tenth the data. Furthermore, LLM-ACES's feedback-driven data acquisition makes it robust to noise and recovers the correct symbolic structure, while baselines introduce spurious terms that fit the data locally but obscure the true governing relationships.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces LLM-ACES, a closed-loop framework for ODE discovery that uses an LLM to propose operator priors partitioning the search space, fits candidate equations within regions, and employs candidate disagreement to drive adaptive trajectory acquisition. On 122 systems from ODEBench and ODEBase it reports the lowest median NMSE (outperforming baselines by several orders of magnitude), symbolic accuracies of 46.2% and 52.4%, improved sample efficiency with one-tenth the data, and greater robustness to noise than baselines.

Significance. If the performance gains prove reproducible and the closed-loop mechanism is shown to be responsible rather than implementation differences, the work would offer a notable advance by demonstrating how LLM-generated priors combined with disagreement-driven active sampling can mitigate identifiability gaps in dynamical-system discovery. The emphasis on sample efficiency and noise robustness addresses practical constraints in experimental settings.

major comments (2)
  1. [Abstract] Abstract: the central empirical claim of orders-of-magnitude lower median NMSE and 46–52% symbolic accuracy on 122 systems is presented without any description of the experimental protocol, baseline implementations, statistical tests, data-generation details, or selection criteria for the 122 systems, rendering the quantitative results unverifiable.
  2. [Abstract] Abstract: the mechanism that LLM-proposed operator priors create distinct regions and that candidate disagreement reliably selects informative trajectories is asserted as the source of the performance gap, yet no ablation is described that holds the acquisition strategy fixed while removing the LLM prior step, nor any quantitative correlation between disagreement and reduction in posterior entropy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to improve clarity and provide additional supporting analyses.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central empirical claim of orders-of-magnitude lower median NMSE and 46–52% symbolic accuracy on 122 systems is presented without any description of the experimental protocol, baseline implementations, statistical tests, data-generation details, or selection criteria for the 122 systems, rendering the quantitative results unverifiable.

    Authors: We agree that the abstract's brevity omits these details. The full manuscript describes the experimental protocol, baseline implementations, statistical tests, data-generation process, and selection criteria for the 122 systems in Sections 4.1, 4.2, and 5. We will revise the abstract to include a concise summary of the experimental setup and direct readers to the relevant sections, thereby improving verifiability while respecting length limits. revision: yes

  2. Referee: [Abstract] Abstract: the mechanism that LLM-proposed operator priors create distinct regions and that candidate disagreement reliably selects informative trajectories is asserted as the source of the performance gap, yet no ablation is described that holds the acquisition strategy fixed while removing the LLM prior step, nor any quantitative correlation between disagreement and reduction in posterior entropy.

    Authors: Section 6 of the manuscript includes ablations examining the LLM priors and disagreement-driven acquisition. However, we acknowledge that a targeted ablation holding the acquisition strategy fixed while removing only the LLM prior step is not explicitly presented, nor is a direct quantitative correlation between disagreement and posterior entropy reduction. We will add this specific ablation and the requested correlation analysis (including a new figure) in the revised manuscript to better substantiate the claimed mechanism. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical framework evaluated on external benchmarks

full rationale

The paper describes LLM-ACES as a procedural framework that uses an LLM to propose operator priors, fits candidate equations within partitioned regions, and employs candidate disagreement to drive adaptive trajectory acquisition. All reported results consist of comparative performance metrics (median NMSE, symbolic accuracy) measured on the fixed external benchmark collections ODEBench and ODEBase across 122 systems. No equation, theorem, or performance claim is shown to reduce by construction to a quantity defined inside the method itself; the evaluation remains independent of the fitted parameters or LLM proposals. Any self-citations that may exist are peripheral and do not carry the central empirical claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities beyond the high-level method description.

axioms (1)
  • domain assumption LLM-generated operator priors can usefully partition the space of possible ODEs
    Invoked as the first step of the method in the abstract.
invented entities (1)
  • LLM-ACES framework no independent evidence
    purpose: Closed-loop ODE discovery via LLM priors and adaptive acquisition
    New method introduced in the paper

pith-pipeline@v0.9.1-grok · 5831 in / 1290 out tokens · 24856 ms · 2026-06-25T23:33:18.597855+00:00 · methodology

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