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arxiv: 2605.07323 · v1 · submitted 2026-05-08 · 💻 cs.AI · cs.LG· cs.NE· cs.SC

Recognition: 2 theorem links

· Lean Theorem

Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation

Sum Kyun Song , Bong Gyun Shin , Jae Yong Lee
Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:59 UTC · model grok-4.3

classification 💻 cs.AI cs.LGcs.NEcs.SC
keywords ordinary differential equationsequation discoveryLLM evaluationmulti-agent systemsymbolic regressionscientific machine learningqualitative assessment
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0 comments X

The pith

DoLQ recovers true ordinary differential equations from data more successfully by using an LLM to check both numerical fit and physical plausibility.

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

The paper introduces DoLQ as a multi-agent system for finding the differential equations that govern observed data. One agent proposes possible equation forms, another tunes their numerical parameters to match the data, and a third agent uses a large language model to judge both how accurately the equation fits the numbers and whether it aligns with physical domain knowledge. This dual qualitative-quantitative loop is repeated to steer the search. On standard benchmarks with multiple interacting variables, the method recovers the exact ground-truth equations at higher rates and with more precise term selection than approaches that use only numerical error metrics.

Core claim

DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations.

What carries the argument

The Scientist Agent, which uses an LLM to evaluate physical plausibility and domain knowledge alongside quantitative accuracy, then synthesizes the two judgments to direct the next round of candidate proposals.

If this is right

  • Higher success rates on multi-dimensional ODE benchmarks than quantitative-only methods.
  • More accurate recovery of the exact symbolic terms present in the ground-truth equations.
  • Incorporation of domain knowledge to favor physically plausible equations even when multiple forms fit the data numerically.
  • Iterative refinement of the equation set through combined qualitative and quantitative feedback.

Where Pith is reading between the lines

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

  • The same multi-agent pattern could be tested on partial differential equations or discrete dynamical systems where qualitative constraints are also important.
  • Performance may degrade if the underlying LLM lacks relevant scientific knowledge or changes between runs, pointing to a need for prompt stabilization or ensemble evaluation.
  • This hybrid setup suggests a broader route for scientific machine learning in which language-model reasoning enforces consistency with known laws while data fitting supplies the coefficients.

Load-bearing premise

The LLM Scientist Agent can reliably judge physical plausibility and domain knowledge without hallucinations, training-data biases, or inconsistent judgments across repeated runs.

What would settle it

Running the Scientist Agent multiple times on identical candidate equations and checking whether its qualitative plausibility scores and term selections remain stable, or presenting a data-fitting but physically invalid equation and verifying that the agent consistently rejects it.

Figures

Figures reproduced from arXiv: 2605.07323 by Bong Gyun Shin, Jae Yong Lee, Sum Kyun Song.

Figure 1
Figure 1. Figure 1: Similar MSE does not imply correct discovery: even with a low MSE, the identified equation may differ from the true one, suggesting that quantitative metrics alone are insufficient and that qualitative evaluation is therefore necessary. in many real-world systems, the form of governing equa￾tions is not clearly known, and one must infer them from observational data. Consequently, there is a growing need fo… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the DoLQ framework for LLM-based ODE discovery. The framework operates through an iterative loop among three components: (1) the Sampler Agent proposes candidate terms with physical justifications based on the system description and Scientist Agent’s feedback; (2) the Parameter Optimizer makes functions and fits their parameters; and (3) the Scientist Agent evaluates each term through qualitati… view at source ↗
Figure 3
Figure 3. Figure 3: Equation comparison on the 2D dimensionless Glider system. Ground truth terms are highlighted in green. Group (b) shows models with reasonable NMSE: DoLQ achieves integral NMSE < 10−3 across all dimensions, and LLM-SR exhibits the lowest NMSE among the remaining methods. Group (c) shows models with higher NMSE. Each model’s discovered equation terms and their corresponding optimized coefficients are displa… view at source ↗
Figure 4
Figure 4. Figure 4: Success scores measured across eight benchmark ODEs. NMSE test: success is achieved if the integral NMSE across all dimensions is < 10−3 . Term test: success is achieved if the discov￾ered equation matches the ground truth structure after excluding terms with negligible impact [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative token usage on the 4D Glider system. Stars (*) indicate that LLM-SR and DoLQ successfully identified the governing equation. NMSE values reach their minimum on the Glider system. and interpretation of the discovered terms compared to the full 4D system. Qualitative examination reveals critical differences in the discovered physical laws. In [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Impact of the Scientist Agent on search convergence for the Glider(2D) problem. DoLQ with the Scientist Agent discovers the correct equation at iteration 27, while the baseline without the Scientist Agent finds it at iteration 62, demonstrating that qualitative feedback accelerates convergence [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Action frequencies for the top-ranked terms in each tar￾get dimension during DoLQ execution on the Glider(2D) problem. Highlighted terms appear in the ground-truth equation and con￾sistently receive high keep frequency, indicating that the Scientist Agent effectively preserves physically meaningful terms. 5. Component validation 5.1. Necessity of the Scientist Agent To demonstrate the critical role of qual… view at source ↗
Figure 8
Figure 8. Figure 8: BFGS alone can fail even with a structurally correct skeleton, whereas the hybrid optimizer successfully recovers the correct solution. This illustrates why hybrid parameter optimiza￾tion is necessary. To further understand how the Scientist Agent operates within DoLQ, we analyze its decision patterns during the discovery process [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: State trajectories for benchmark ODEs in ID and ID-Ext regimes. The figures display: ID 1: SIR infection model; ID 2: Glider (dimensionless); ID 3: Reduced model for chlorine dioxide-iodine-malonic acid reaction; ID 4: Isothermal autocatalytic reaction model; ID 5: Interacting bar magnets; ID 6: Binocular rivalry model; ID 7: Oscillator death model; ID 8: Glider (physical units). 15 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 10
Figure 10. Figure 10: Visual representation of the function construction process. A list of symbolic terms (left) is converted into an executable Python function (right) with learnable coefficients (params) and a bias term, enabling numerical optimization. C. Comprehensive experimental results C.1. Discovered governing equations In this section, we present the explicit mathematical forms of the governing equations discovered b… view at source ↗
Figure 11
Figure 11. Figure 11: Adoption frequency of BFGS and differential evolution across different ODE systems. Differential evolution is more frequently selected for systems with intricate functional forms where the loss landscape is more rugged, while BFGS is often sufficient for simpler systems. E. Quantitative results under shifted initial conditions To evaluate robustness against initial-state variations, the discovered equatio… view at source ↗
Figure 16
Figure 16. Figure 16: shows the initialization-time Sampler prompt used before any prior evaluations are available, making explicit what context is given when accumulated knowledge, term-level feedback, and removed-term history are still empty. Initial Sampler Prompt (Iteration 1) You are a helpful assistant tasked with discovering mathematical term structures for scientific systems. Complete the term list below, considering p… view at source ↗
Figure 12
Figure 12. Figure 12: The Sampler Prompt. The labeled sections (1a)-(1c) correspond to the Sampler components in [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example response from the Sampler LLM showing the structured JSON response containing candidate ODE terms for each dimension. Scientist Prompt (2a) You are a senior scientist specializing in ODE discovery. Your role is to evaluate proposed mathematical terms and provide guidance to improve the term list in the next iteration. Progress: Currently on iteration 50 of 100 total System Description: Glider flig… view at source ↗
Figure 14
Figure 14. Figure 14: The Scientist Prompt. The labeled sections (2a)-(2d) correspond to the Scientist components in [PITH_FULL_IMAGE:figures/full_fig_p036_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Example response from the Scientist LLM showing term-by-term evaluation with semantic quality assessment and action recommendations. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗
read the original abstract

Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential equation modeling also requires incorporating domain knowledge to ensure physical plausibility. To address this gap, we propose DoLQ, a method for discovering ordinary differential equations with LLM-based qualitative and quantitative evaluation. DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations. Our code is available at https://github.com/Bon99yun/DoLQ.

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

3 major / 2 minor

Summary. The paper proposes DoLQ, a multi-agent architecture for ODE discovery from data consisting of a Sampler Agent for candidate generation, a Parameter Optimizer for fitting, and an LLM-powered Scientist Agent that performs qualitative evaluation of physical plausibility/domain knowledge alongside quantitative metrics to iteratively steer the search. Experiments on multi-dimensional ODE benchmarks are reported to show higher success rates and more accurate recovery of ground-truth symbolic terms than prior methods, with code released at https://github.com/Bon99yun/DoLQ.

Significance. If the performance gains hold under rigorous controls, the work would be significant for scientific machine learning by showing how LLM-based qualitative assessment can be integrated into symbolic regression pipelines to improve physical plausibility of discovered ODEs. The open-source code is a clear strength for reproducibility.

major comments (3)
  1. [Method] Method section (Scientist Agent description): the qualitative evaluation procedure supplies no prompt templates, temperature settings, consistency metrics across stochastic calls, or human-expert agreement studies; because the iterative guidance and claimed gains rest on these LLM judgments being stable and unbiased, this omission is load-bearing for the superiority claim over quantitative-only baselines.
  2. [Experiments] Experiments section: the headline results on success rates and symbolic-term recovery do not report the number of independent trials per benchmark, statistical significance tests, exact baseline implementations, or explicit handling of LLM stochasticity; without these, the gap between claim and verifiable evidence remains moderate.
  3. [Results] Results/discussion: the advantage attributed to the multi-agent design with LLM qualitative synthesis could arise from benchmark-specific LLM behavior rather than the architecture itself, given the absence of bias controls or ablation isolating the Scientist Agent's contribution.
minor comments (2)
  1. The abstract refers to 'multi-dimensional ordinary differential equation benchmarks' without enumerating the specific systems or providing a table of their dimensions and ground-truth forms.
  2. [Method] Notation for the combined qualitative-quantitative score synthesized by the Scientist Agent is introduced without an explicit equation or pseudocode step.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We appreciate the emphasis on reproducibility, statistical rigor, and isolating the contributions of our multi-agent design. Below we respond point-by-point to the major comments and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Method] Method section (Scientist Agent description): the qualitative evaluation procedure supplies no prompt templates, temperature settings, consistency metrics across stochastic calls, or human-expert agreement studies; because the iterative guidance and claimed gains rest on these LLM judgments being stable and unbiased, this omission is load-bearing for the superiority claim over quantitative-only baselines.

    Authors: We agree that full transparency on the Scientist Agent is essential. In the revised manuscript we will add the complete prompt templates (both for qualitative physical-plausibility assessment and for synthesizing quantitative/qualitative scores), state the temperature (0.7) and other generation parameters, and report consistency metrics obtained by repeating each LLM call three times with different seeds and measuring agreement on the final recommendation. We did not conduct a formal human-expert agreement study in the original work; we will therefore add an explicit limitations paragraph acknowledging this gap and noting that LLM judgments may carry domain-specific biases. These changes directly address the load-bearing concern by allowing readers to reproduce and scrutinize the qualitative component. revision: partial

  2. Referee: [Experiments] Experiments section: the headline results on success rates and symbolic-term recovery do not report the number of independent trials per benchmark, statistical significance tests, exact baseline implementations, or explicit handling of LLM stochasticity; without these, the gap between claim and verifiable evidence remains moderate.

    Authors: We accept this criticism. The revised Experiments section will explicitly state that all reported figures are means over five independent trials per benchmark, include statistical significance tests (paired t-tests and Wilcoxon signed-rank tests with p-values) comparing DoLQ against each baseline, provide the precise code versions and hyper-parameters used for every baseline (with links to our re-implementations), and describe our handling of LLM stochasticity via fixed random seeds plus reporting of standard deviations across runs. These additions will close the gap between claims and verifiable evidence. revision: yes

  3. Referee: [Results] Results/discussion: the advantage attributed to the multi-agent design with LLM qualitative synthesis could arise from benchmark-specific LLM behavior rather than the architecture itself, given the absence of bias controls or ablation isolating the Scientist Agent's contribution.

    Authors: We recognize the need to isolate the Scientist Agent's contribution. In the revision we will add a dedicated ablation study that runs the full DoLQ pipeline against an otherwise identical quantitative-only variant (i.e., Sampler + Parameter Optimizer without the LLM qualitative synthesis step). We will also expand the benchmark suite and include a short discussion of potential LLM biases together with controls (e.g., temperature sweeps and prompt-variation checks). These new results will allow readers to assess whether the observed gains are attributable to the multi-agent architecture rather than benchmark-specific LLM behavior. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical multi-agent method with external LLM judgments and benchmark validation

full rationale

The paper describes an algorithmic framework (Sampler, Optimizer, Scientist Agent) that uses an LLM for qualitative evaluation of physical plausibility and combines it with quantitative metrics to guide iterative search for ODEs. Performance is assessed via success rates and symbolic recovery on external multi-dimensional ODE benchmarks, with no mathematical derivation chain, fitted parameters renamed as predictions, or self-citations that bear the central claim. The LLM judgments are treated as an independent external input rather than derived from the method's own outputs, and the architecture does not reduce any claimed result to a quantity defined in terms of its own fitted values or prior self-referential theorems. This is a standard empirical proposal whose validity rests on experimental outcomes outside the paper's internal definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the method implicitly relies on LLM prompt engineering and evaluation criteria whose details are not stated.

pith-pipeline@v0.9.0 · 5456 in / 1174 out tokens · 23877 ms · 2026-05-11T00:59:45.676016+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

30 extracted references · 11 canonical work pages · 1 internal anchor

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    Found” columns show the cumulative tokens consumed up to the point when the method successfully discovered a reasonable equation (if applicable), while “Total

    + (0.0004499464651382155∗x2) + (0.0002873146658744879∗x3) + (3.203979516768474e−05∗x0∗x1) ˙x2 (−(−8.313688998942643e−05)∗sin(x1)) + (0.00012926385963329574∗ −0.0002110727122192323∗x3∗x0 2 ∗sin(x1)) + (−0.00012926385963329574∗ 0.05∗x0 2 ∗cos(x1)) + (−0.00012926385963329574∗1.5157577205870516∗ (−0.0002110727122192323∗x3) 2 ∗x0 2 ∗cos(x1))+(−1.0∗exp(−((x3−1....

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  19. [20]

    -- This system is 2-dimensional -- Variables x2 and above do not exist

    Target System Context: Input variables are x0, x1. -- This system is 2-dimensional -- Variables x2 and above do not exist

  20. [21]

    x0", "np.sin(x0)

    Term Format: Propose terms WITHOUT coefficients. The system will automatically attach trainable parameters. -- Correct: "x0", "np.sin(x0)", "x0 *x1" -- Incorrect: "params[0] *x0", "C *x0", "0.5 *x0"

  21. [22]

    np.sin(2 *x0)

    Term Complexity: You MAY use internal constants if they have physical meaning (e.g., frequency, phase). -- Example: "np.sin(2 *x0)" is allowed and encouraged if the factor 2 is significant. -- Note: The system will still attach an outer trainable parameter (e.g., params[0] *np.sin(2*x0))

  22. [23]

    9.81 *x0

    Symbolic Constants: Do NOT use symbolic constants like ’g’, ’k’, ’m’. Use numerical values. -- Correct: "9.81 *x0" (if g=9.81 is known), "np.pi *x0" -- Incorrect: "g *x0" (will cause NameError)

  23. [25]

    x0", "x1

    Reasoning required: When proposing each term, provide a physical/mathematical reasoning based on the system description (desc). [Example (2D System)] x0 t: ["x0", "x1"] x1 t: ["x0", "np.sin(x1)"] Figure 16.Initial prompt to the Sampler agent at iteration 1, where accumulated knowledge, term-level evaluation, and removed-term history are empty at initializ...

  24. [26]

    You can use: import numpy as np

  25. [27]

    - This system is 4-dimensional - Variables x4 and above do not exist

    Target System Context: Input variables are x0, x1, x2, x3. - This system is 4-dimensional - Variables x4 and above do not exist

  26. [28]

    x0", "np.sin(x0)

    Term Format: Propose terms WITHOUT coefficients. The system will automatically attach trainable parameters. - Correct: "x0", "np.sin(x0)", "x0 *x1" - Incorrect: "params[0] *x0", "C *x0", "0.5 *x0"

  27. [29]

    np.sin(2 *x0)

    Term Complexity: You MAY use internal constants if they have physical meaning (e.g., frequency, phase). - Example: "np.sin(2 *x0)" is allowed and encouraged if the factor 2 is significant. - Note: The system will still attach an outer trainable parameter (e.g., params[0] *np.sin(2*x0))

  28. [30]

    9.81 *x0

    Symbolic Constants: Do NOT use symbolic constants like ’g’, ’k’, ’m’. Use numerical values. - Correct: "9.81 *x0" (if g=9.81 is known), "np.pi *x0" - Incorrect: "g *x0" (will cause NameError)

  29. [31]

    Structural modifications are required

    No duplicates: Equations identical to previous attempts are forbidden. Structural modifications are required

  30. [32]

    x0", "x1 *x2

    Reasoning required: When proposing each term, provide a physical/mathematical reasoning based on the system description (desc). [Example (4D System)] x0 t: ["x0", "x1 *x2", "x3"] x1 t: ["x0", "np.sin(x1)"] x2 t: ["x0 *x1"] x3 t: ["-9.81", "x0"] Figure 12.The Sampler Prompt. The labeled sections(1a)-(1c)correspond to the Sampler components in Figure 2:(1a)...