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Overview of all projectsLLM-AutoSciLab: Closed-Loop Scientific Discovery via Active Experimentation with LLMs
Couples LLM hypothesis generation with hypothesis-conditioned experiment selection, and introduces the ActiveSciBench benchmark for active discovery.
NeurIPS 2026LLEMA: Evolutionary Search with LLMs for Multi-Objective Materials Discovery
Couples the scientific knowledge in LLMs with chemistry-informed evolutionary rules and memory-based refinement for materials discovery.
ICLR 2026LLM-FE: Automated Feature Engineering with Large Language Models
A framework that casts feature engineering as evolutionary program search, with LLMs generating and refining executable feature transformations.
TMLR 2026
NeurIPS 2026
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.
Why Passive Data Is Not Enough
When training trajectories cover only part of the state space, structurally different equations fit the data equally well. The paper illustrates this with dx/dt = x + 0.1x³: sampled at small x, the cubic term is invisible, so passive methods return a linear law with negligible training error. Hover the plot to see where the candidates part ways.
The identifiability gap, from the paper’s Appendix D.2
Right-hand side f(x) of the recovered equations, log–log axes. Inside the shaded region every candidate matches the data; outside it the passive fits miss the cubic growth.
Recovered equations
Ground truth: x + 0.1x³. LLM-ODE’s fit is negative on this range, so it is not drawn on the log axes.
| Method | Reconstructed equation |
|---|---|
| SINDy / E-SINDy | 1.0096x |
| PySR | x |
| Operon | 1.023x + 0.0038 |
| ODEFormer | 1.0053x |
| LLM-ODE | 0.0001x − 0.261 |
| LLM-ACES | x + 0.1x3 |
“No amount of optimization on the same trajectory can reveal structure that is absent from the data.” LLM-ACES instead queries new initial conditions where its candidate equations disagree most, which is exactly where the cubic term shows itself.
Method Overview
LLM-ACES treats discovery as a closed loop in which the symbolic hypothesis space and the acquired dataset co-evolve. The LLM never writes the final equation; it shapes the space that symbolic regression searches, and the fitted candidates decide which experiment to run next.
LLM-guided hypothesis construction
Each round the LLM proposes K = 3 operator priors (sets of unary and binary operators): one exploits the best patterns in the experience buffer, the others explore operator families not yet tried.
Hypothesis generation
PySR fits a candidate equation for every state dimension inside each constrained subspace. The accumulated population is a structured picture of what the dynamics could be.
Active experiment selection
Every candidate is rolled out from a pool of initial conditions; the oracle is queried where the average pairwise NMSE between rollouts, the predictive divergence, is largest.
Evaluation and feedback
New trajectories are split into train and validation data and all candidates are re-scored. The two best and two worst per dimension become demonstrations and failure cases in the next prompt.
Main Results
Twelve methods on ODEBench (63 physics-inspired systems) and ODEBase (59 biological systems), evaluated on the training trajectory, a held-out initial condition, and a 10× longer horizon. All LLM methods see anonymized variables.
Data fidelity: median NMSE per method
Each dot is a method’s median NMSE across the benchmark’s systems, on a log scale where further left is better. The bracket measures the gap between the best LLM-ACES variant and the best baseline.
Show the full results table
Structural recovery: right equation, right size
Symbolic accuracy (share of equations whose structure matches the ground truth) against mean expression complexity. Methods that only fit the data drift away from the ground-truth complexity line or sit near zero accuracy. Hover a point to see the method.
Analysis
Which components matter, how much data LLM-ACES needs, and why the benchmarks are anonymized.
Ablation: every component contributes
NMSE distributions on 15 stratified ODEBench systems (5 each of 1D, 2D and 3D). Boxes span the interquartile range, the dark tick is the median, circles are outlying systems. Dropping the LLM priors hurts most; replacing predictive divergence with random initial conditions hurts generalization and OOD.
Exactly recovered systems have zero error and are shown at the 10−35 floor used in the paper’s figure.
Show data table
Sample efficiency
Reconstruction error as the observation budget grows. The blue rule marks what LLM-ACES reaches with 100 samples; no baseline gets below it even with 10× the data.
Show data table
Memorization check
Qwen3-32B refining equations for 100 iterations. With the original variable names it reaches near-perfect fits on Feynman and ODEBench within a few iterations; once the variables are anonymized it stalls near 10−4 and exact recoveries collapse from 17 to 2.
Equations recovered exactly
Show data table
Qualitative Results
Final equations for the Schnackenberg and Maxwell–Bloch systems. Green marks components that match the ground truth; red marks spurious terms.
Passive regression oversimplifies
PySR and SINDy return sparse or nearly linear expressions that drop the essential nonlinear interaction terms, even with low error on the observed trajectory.
Richer searches invent terms
Operon and ODEFormer add spurious polynomial or oscillatory terms (ODEFormer even inserts a sine) that improve local interpolation while obscuring the mechanism.
Acquisition alone is not identifiability
LLM-ODE recovers some components but misses coupled interactions, and APPS-ODE acquires data yet still returns oversimplified equations: the queried trajectories must separate competing hypotheses.
LLM-ACES recovers the mechanism
Querying where candidate equations disagree, inside LLM-shaped operator spaces, yields compact and correct equations, including the reaction term of Schnackenberg and the bilinear couplings of Maxwell–Bloch.
More from the Appendix
Full distributions, robustness to noise and sparse sampling, the effect of how trajectories are chosen, and LLM-ACES rollouts for every benchmark system.
BibTeX
@inproceedings{abhyankar2026llmaces,
title={{LLM}-{ACES}: Closed-Loop Discovery of Dynamic Systems with {LLM}-Guided Adaptive Search},
author={Abhyankar, Nikhil and Li, Sha and Kabra, Sanchit and Ramakrishnan, Naren and Gel, Yulia and Reddy, Chandan K},
booktitle={The Fortieth Annual Conference on Neural Information Processing Systems (NeurIPS)},
year={2026},
}