Learning from Less: SINDy Surrogates in RL

Aniket Dixit; Faizan Ahmed; James Brusey; Muhammad Ibrahim Khan

REVIEW 4 major objections 6 minor 1 cited by

Learning from Less: SINDy Surrogates in RL

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read With as few as 75 environment transitions, the SINDy algorithm fits a surrogate RL environment whose state predictions correlate with the true physics engine at 0.997 or better, and agents trained inside that surrogate match…

desk verdict SINDy surrogates are a plausible idea, but this paper never tests whether policies trained in them actually perform in the original environments, so its headline efficiency claims are unsupported. read the letter →

arxiv 2504.18113 v1 pith:A44E3PEK submitted 2025-04-25 cs.LG cs.AI

Aniket Dixit , Muhammad Ibrahim Khan , Faizan Ahmed , James Brusey This is my paper · ORCID

classification cs.LGcs.AI

keywords model-basedreinforcementlearningSINDysparseidentificationofnonlineardynamicssurrogateenvironmentsample-efficientRLMountainCarLunarLanderinterpretable

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that SINDy, the sparse-identification method for nonlinear dynamics, can build a surrogate reinforcement-learning environment from a tiny number of recorded transitions and still reproduce the original dynamics faithfully. Specifically, it reports that 75 transitions from one episode suffice for Mountain Car and 1,000 for Lunar Lander, giving state-wise next-state correlations between 0.99 and 1.00 and mean squared errors down to $3.11\times10^{-6}$. If that holds, agents can be trained mostly inside a cheap analytical model rather than a physics engine, cutting total training steps by 20–35% while yielding comparable final policies and leaving the engineer explicit equations to inspect.

What carries the argument

The engine of the paper is the SINDy surrogate: a sparse regression model that selects a few terms from a library of candidate functions—polynomials, trigonometric terms, and rational functions—via Sequential Thresholded Least Squares, yielding an explicit transition rule $s_{t+1}=f_{SINDy}(s_t,a_t)$. The authors wrap that learned rule in the same RL environment interface as the original benchmarks so any policy algorithm can train against it in place of the physics engine. The fitting procedure that carries the argument is a four-step loop: initial STLSQ fit, residual-driven addition of nonlinear terms, grid search over sparsity threshold and regularization strength, and cross-validated selection by mean squared error. The paper's key modeling lesson is that the candidate library must be matched to the dynamics: trigonometric functions are needed for Mountain Car's oscillatory behavior, while polynomials suffice for Lunar Lander.

What would settle it

Take the same SINDy pipeline but collect the 75 Mountain Car transitions from a trajectory that never enters, or only barely enters, the goal region; if the fitted surrogate's held-out predictions over the full state space drop below 0.99 correlation, the single-episode coverage assumption is broken. Separately, take the surrogate-trained Lunar Lander policy and run it in the original environment: if its success rate is materially below a natively trained policy, the policy-transfer claim is falsified.

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Extended reading notes

Core claim

The discovery the authors argue for is that sparse dynamics identification generalizes from a single trajectory to the full state-transition function well enough to support policy learning. Fitting SINDy to 75 Mountain Car transitions, collected under an $\epsilon$-greedy version of a pretrained Soft Actor-Critic policy, and to 1,000 Lunar Lander transitions, collected under a pretrained Proximal Policy Optimization agent, produced explicit maps $s_{t+1}=f_{SINDy}(s_t,a_t)$ whose held-out state predictions correlate with the true simulator at 0.997–1.000 for most variables; the reported worst case is 0.989 for Lunar Lander angular velocity, and the mean squared errors reach $3.11\times10^{-6}$ for Mountain Car velocity and $1.42\times10^{-6}$ for Lunar Lander position. Agents trained in the surrogates converged much like agents trained in the original environments, and the surrogate-trained runs needed 65,075 steps versus 100,000 for Mountain Car and 801,000 versus 1,000,000 for Lunar Lander.

Load-bearing premise

The whole argument rests on the assumption that the 75 Mountain Car or 1,000 Lunar Lander transitions, collected from a single episode under an $\epsilon$-greedy pretrained policy, are representative enough of the full state-action distribution that SINDy's sparse equation generalizes to every region a trained policy will visit.

Editorial extensions

If this is right

Because surrogate rollouts come from an explicit formula rather than a physics engine, model-based RL becomes practical when interaction budgets are tiny, such as physical robots or safety-critical control tasks.
Agents can be pre-trained in the surrogate and then fine-tuned in the original environment, since simulating the sparse equations is nearly free once the model is learned.
The explicit governing equations let engineers audit what the agent is modeling, debug failure modes, and check safety properties in ways a neural-network surrogate does not easily allow.
The library-selection result gives practitioners a usable heuristic: include trigonometric basis functions when the environment has oscillatory dynamics, and otherwise prefer low-order polynomial terms.
The measured 20% (Lunar Lander) and 35% (Mountain Car) reductions in total training steps give a concrete expectation for the computational savings a sparse surrogate can deliver.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The validity of the 75-transition fit is tied to the single episode used for collection; in a new task, one would need to verify that the sampled trajectory actually covers the state-action region the final policy visits, since SINDy only fits within the library functions it selects.
A natural test the paper leaves implicit is to deploy the surrogate-trained policies directly in the original environments and measure return; the 20–35% step savings only translate to real systems if the surrogate-to-true gap stays small enough not to break the transferred policy.
The data requirement could probably be pushed further by active collection: use the SINDy model's current prediction error to ask for the next transition where the model is most uncertain, rather than relying on one fixed exploratory episode.
The reported worst-case correlation of 0.989 for Lunar Lander angular velocity suggests that aggregate correlation can hide a weak state; a stricter per-component floor, with library or threshold selection targeted at the weakest variable, would be a cheap robustness improvement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: the paper's headline claim — that SINDy surrogates give comparable RL policies from 75–1000 interactions — is not backed by the evidence it presents. There is no transfer evaluation of surrogate-trained policies back into the original environments. All we get is fit quality on held-out transitions and visual policy plots.

What's genuinely useful: the paper shows that with a well-chosen feature library (trigonometric for Mountain Car, polynomial for Lunar Lander), SINDy can fit the dynamics of these two gym environments from a single episode of a pretrained policy. The MSE values (10^-6) are believable, and the comparison of library functions is a nice empirical detail. The interpretable equations are a real plus over neural surrogates.

Soft spots: first, the central claim of comparable performance is unsupported. Section 3.1 has no return table, no success-rate comparison, no error bars for policies trained in the surrogate and evaluated in the original environment. The paper only compares policy maps inside each environment. The abstract promises 'comparable performance' but the evidence is visual. That's a load-bearing gap. Second, the novelty claim is wrong: the paper cites Arora et al. (2022) and Zolman et al. (2024), both doing SINDy-based RL surrogates, then claims 'first application' in the introduction. That contradiction suggests carelessness at best. Third, the data collection is one epsilon-greedy episode from a pretrained agent; the fitted dynamics are only validated on held-out transitions from that same trajectory. RL training in the surrogate will explore off-trajectory states, and the paper does not show the model is accurate there. Section 4 concedes exactly this: 'generalization capability to significantly different initial conditions remains to be fully explored.' So the efficiency claim is real only if the surrogate dynamics are reliable for exploration, which is untested.

Who is this for? Researchers working on model-based RL who want a concrete example of SINDy applied to simple benchmarks. The paper's value is as a proof-of-concept, not a validated method. It deserves a serious referee: the question is interesting and the missing transfer evaluation is fixable with a straightforward experiment — return and success-rate tables for surrogate-trained policies in the original environments, with error bars. But in current form it should not be accepted.

I'd send it to review, but I would not accept it without those numbers.

Referee Report

4 major / 6 minor

Summary. The paper proposes using Sparse Identification of Nonlinear Dynamics (SINDy) to learn surrogate models of two OpenAI Gym environments, Mountain Car and Lunar Lander, from very small datasets (75 and 1,000 transitions, respectively), and then training RL agents within these surrogates. It reports high state-prediction accuracy (correlations above 0.99) and claims that agents trained in the surrogates achieve comparable performance to agents trained in the original environments while using 20-35% fewer training steps. The manuscript also discusses the importance of the feature library (trigonometric vs. polynomial terms) and the interpretability benefits of SINDy.

Significance. If the claims were substantiated, the work would offer a data-efficient and interpretable alternative to neural-network surrogate models in model-based RL, with potential practical value. The paper's pipeline is clearly described and uses standard benchmarks, and the reported dynamics-model accuracy is promising. However, the central claim of 'comparable agent performance' is never quantitatively tested: the manuscript's main evidence consists of visual policy heatmaps and training-step counts. The paper also credits itself with a 'systematic evaluation framework' that is not actually present in the reported experiments. As a result, the significance of the contribution cannot be assessed from the current text, and the empirical core needs substantial strengthening.

major comments (4)

[Section 3.1 and Abstract] The central claim that RL agents trained in the surrogate environments achieve comparable performance to agents trained in the original environments is not supported by any quantitative evaluation. The paper reports only per-variable MSE and correlation for the SINDy model (Table 1) and visual policy heatmaps (Figures 1 and 2); it does not report cumulative rewards, success rates, or final performance metrics for surrogate-trained policies evaluated in the original Mountain Car or Lunar Lander environments. Consequently, the abstract's assertions of 'comparable performance' and 'similar ... final performance metrics' are unverified. To support the central claim, the authors must add quantitative results, including return curves or success rates, for policies trained in the surrogate and evaluated in the original environment, with multiple seeds and error bars.
[Section 2.1 and Section 4] The data collection strategy uses a single episode from a pre-trained epsilon-greedy policy (75 transitions for Mountain Car, 1,000 for Lunar Lander). This does not establish that the learned SINDy model is accurate on the state-action distribution visited by an RL agent trained in the surrogate, which will explore regions beyond the single training trajectory and may exploit model errors. The paper's own limitation statement in Section 4 ('generalization capability to significantly different initial conditions remains to be fully explored') acknowledges this gap. The authors should provide evidence of state-space coverage (e.g., visitation histograms or rollout error on states reached during surrogate training) or collect data from a more exploratory policy.
[Section 3.2] The 'fewer total steps' comparison (65,075 vs. 100,000 for Mountain Car; 801,000 vs. 1,000,000 for Lunar Lander) is not a valid measure of computational cost reduction as claimed in the abstract ('reducing computational costs by 20-35%'). These numbers compare training steps in the surrogate versus the original environment, but they ignore the 75/1,000 real-environment interactions needed for data collection and the cost of training the SINDy model. Moreover, the paper does not report wall-clock time or any hardware-independent compute metric. The claim of reduced computational cost must be either redefined as 'fewer surrogate training steps' or supported by actual computational measurements.
[Section 3.2] The comparison to a neural-network surrogate (MSE 3.11e-6 vs. 4.45e-6; approximately 95% less computational resources) is not reproducible because the neural network architecture, training procedure, and evaluation protocol are not described anywhere in the paper. Either provide complete experimental details for this baseline or remove the comparison and its associated efficiency claim.

minor comments (6)

[Table 1] In the lower part of Table 1, the columns 'MC MSE' and 'LL MSE' do not specify which state variable (e.g., position or velocity) the mean squared error is computed over, which makes the library-function comparison difficult to interpret. Please clarify.
[Figure 2] The subplot labels in Figure 2 are duplicated: both the top and bottom rows use labels (a), (b), (c). Rename the six panels as (a)-(f) to avoid confusion.
[Algorithm 1] The pseudocode in Algorithm 1 loops over N episodes for data collection, but the experiments in Section 2.1 use a single episode (N=1). Please make the pseudocode consistent with the reported experiments or clarify that N=1 in the current study.
[Introduction] The claim that 'This work represents the first application of SINDy to RL environment modeling' is contradicted by the cited references, e.g., Zolman et al. (2024) 'Sindy-rl: Interpretable and efficient model-based reinforcement learning' and Arora et al. (2022). Please either revise the novelty statement to accurately reflect prior work or clarify the specific distinction from these existing methods.
[Abstract] The abstract states 'reducing computational costs by 20-35%,' but the only quantitative evidence in Section 3.2 is a reduction in the number of training steps, not a measure of computational cost. Please use terminology that matches the actual measurements (e.g., 'training steps').
[Section 3.1] The term 'state-wise correlations exceeding 0.997' is not a standard accuracy metric for dynamics models; correlation can be high even with systematic bias. Please report residual statistics or ensure MSE values are provided with uncertainty estimates as well.

Circularity Check

1 steps flagged · score 6.0 of 10

Reported surrogate accuracy is the model-selection criterion rather than an independent prediction; policy-transfer claims are untested but not circular.

fitted input called prediction [Section 2.2 'Cross-Validation' and Table 1 / Abstract]
"Cross-Validation: Models were validated using held-out data to ensure generalization performance, with final selection based on minimal MSE. ... we achieve state-wise correlations exceeding 0.997, with mean squared errors as low as 3.11e-06"

The paper reports held-out MSE and correlation values as evidence of SINDy fidelity, but these are exactly the validation metrics used for model selection ('final selection based on minimal MSE'). No separate test set is described, so the reported accuracy is the optimization objective itself: the chosen model minimizes the same loss that is then presented as an independent predictive result. With only 75 or 1000 transitions, this is a fitted input reported as a prediction, not an independent test of the surrogate's generalization.

full rationale

The SINDy modeling itself is not circular: the surrogate is fitted to genuine OpenAI Gym transition data and could, in principle, be validated against independent rollouts or by transferring trained policies back to the original environment. However, the only quantitative evidence of fidelity in the paper is the held-out MSE/correlation used during model selection, with no indication of a separate test set; thus the accuracy claim reduces to reporting the model-selection loss. The downstream RL comparison consists of policy visualizations and convergence descriptions, and no final-return table or original-environment evaluation of surrogate-trained policies is provided, so the 'comparable performance' and 'fewer total steps' claims are unsupported rather than circular. There is no load-bearing self-citation: the cited SINDy-in-RL works are by different authors and are used only for the surrogate-environment integration. Overall, the central accuracy metric is partially circular because it is the fitted selection objective, yielding a score of 6.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper's claims rest on the completeness of the SINDy function library, the representativeness of a single-episode dataset, and the preservation of the original reward function. None of these premises is independently verified, and the threshold/regularization parameters are tuned on the same environments whose dynamics they are meant to capture.

free parameters (5)

STLSQ threshold = not reported
Selected by grid search over sparsity and accuracy in Section 2.2; the actual value is omitted.
STLSQ regularization coefficient = not reported
Grid-searched alongside the threshold; value omitted.
Feature library composition = Mountain Car: polynomial + trigonometric; Lunar Lander: polynomial
Chosen iteratively via residual analysis (Section 2.2, Table 1 bottom); not derived from structure of the systems.
Data collection epsilon = 0.2
Fixed at 0.2 in Section 2.1 to balance exploration and exploitation; no sensitivity analysis.
Number of collected transitions = 75 (Mountain Car), 1,000 (Lunar Lander)
Chosen to demonstrate extreme data efficiency; no rule or calculation given in Section 2.1.

assumptions (4)

domain assumption The SINDy feature library (chosen via residual analysis) contains the true terms of the environment dynamics.
The paper never proves the library is complete; it relies on the fact that Mountain Car has trigonometric terms and Lunar Lander is polynomial (Section 2.2).
domain assumption The reward function from the original environment remains valid when the dynamics are replaced by the SINDy model.
Section 2.3 states the surrogate preserves key characteristics except the state transition; this assumes the reward logic does not depend on physical consistency of the model.
ad hoc to paper A single episode from a pre-trained agent with epsilon-greedy explores enough of the state space to identify dynamics.
No coverage analysis is provided; this is the paper's own data-efficiency bet.
standard math STLSQ sparse regression converges to the true dynamics when data is noiseless and the library is complete.
Standard SINDy assumption from Brunton et al. (2016) cited in Section 1.

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Cite this review

Pith. "Pith review of Learning from Less: SINDy Surrogates in RL." pith.science (2026). https://pith.science/paper/A44E3PEK

@misc{pith2026250418113,
  author       = {Pith},
  title        = {Pith review of: Learning from Less: SINDy Surrogates in RL},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A44E3PEK}},
  note         = {Machine review of arXiv:2504.18113}
}

read the original abstract

This paper introduces an approach for developing surrogate environments in reinforcement learning (RL) using the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm. We demonstrate the effectiveness of our approach through extensive experiments in OpenAI Gym environments, particularly Mountain Car and Lunar Lander. Our results show that SINDy-based surrogate models can accurately capture the underlying dynamics of these environments while reducing computational costs by 20-35%. With only 75 interactions for Mountain Car and 1000 for Lunar Lander, we achieve state-wise correlations exceeding 0.997, with mean squared errors as low as 3.11e-06 for Mountain Car velocity and 1.42e-06 for LunarLander position. RL agents trained in these surrogate environments require fewer total steps (65,075 vs. 100,000 for Mountain Car and 801,000 vs. 1,000,000 for Lunar Lander) while achieving comparable performance to those trained in the original environments, exhibiting similar convergence patterns and final performance metrics. This work contributes to the field of model-based RL by providing an efficient method for generating accurate, interpretable surrogate environments.

Figures

Figures reproduced from arXiv: 2504.18113 by the authors.

**Figure 2.** Lunar Lander policy comparison showing consistent control strategies across both envi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

9 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...
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Model-Based Reinforcement Learning with SINDy

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Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113 0 (15): 0 3932--3937, 2016

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arXiv 2024
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@esa (Ref

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[9]

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[10]

Nathan and Brunton, Steven L

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2018 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

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Model-Based Reinforcement Learning with SINDy

Rishabh Arora, Bruno Castro da Silva, and Eben Moss. Model-based reinforcement learning with sindy. arXiv preprint arXiv:2208.14501, 2022

work page Pith review arXiv 2022

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Discovering governing equations from data by sparse identification of nonlinear dynamical systems

Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113 0 (15): 0 3932--3937, 2016

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Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor

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@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command n...

[8] [9]

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