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REVIEW 3 major objections 5 minor 20 references

Analytical Lyapunov Function Discovery: An RL-based Generative Approach

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A reinforcement-learning-trained transformer writes analytical Lyapunov functions for nonlinear systems up to ten dimensions.

desk verdict The method is genuinely new and well-engineered, but the headline certificates don't satisfy the stated verification as written; the lossy power system claim hinges on an unstated manifold constraint. read the letter →

arxiv 2502.02014 v3 pith:GTFBM7P4 submitted 2025-02-04 cs.LG cs.AIcs.SCcs.SYeess.SY

Haohan Zou , Jie Feng , Hao Zhao , Yuanyuan Shi This is my paper · ORCID
classification cs.LGcs.AIcs.SCcs.SYeess.SY MSC 93D3037N3568T07
keywords Lyapunovfunctionsreinforcementlearningsymbolictransformerformalverificationnonlineardynamicalsystemspowersystemstabilitygeneticprogrammingregression
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 tries to show that a generative transformer, trained from scratch on a single system via reinforcement learning, can produce short symbolic Lyapunov functions for nonlinear dynamics, including non-polynomial and high-dimensional cases where neural-network certificates are hard to verify and sum-of-squares methods struggle. Because the candidates are analytical expressions rather than neural networks, the framework can check them with global optimization during training and certify them with an SMT solver in milliseconds. If true, this turns Lyapunov-function design from a manual art into an automatable search, and it supplies interpretable stability certificates that engineers can inspect. The headline application is a 2-bus lossy power system, for which the paper claims the first analytical local Lyapunov functions in the control literature.

What carries the argument

The carrying object is a symbolic transformer that represents both dynamics and candidate Lyapunov functions as pre-order traversals of expression trees; a decoder samples candidates token by token, conditioned on the dynamics and on parent–sibling tree context. Because candidates are short analytic expressions, SHGO can be run over the state space to find minimizers of $\tilde{V}$ and of $-L_f\tilde{V}$, the natural violation points, and the resulting counterexamples are fed back into the reward set. The reward is the bounded empirical Lyapunov risk $R(\tilde{V})=1/(1+L(\tilde{V}))$, and the risk-seeking policy gradient with $\alpha=0.1$ updates the transformer only on the best quantile of samples; a genetic-programming module further refines candidates into an elite set that drives a supervised guidance loss.

What would settle it

Run dReal with the paper's stated settings ($\epsilon=10^{-3}$, tolerance $\delta=10^{-12}$) over the full box $D=\{(\delta_1,\delta_2,\omega_1,\omega_2): |\delta_i|\le 0.75,\ |\omega_i|\le 2\}$ and query whether $V_1(\delta_1,\delta_2,\omega_1,\omega_2)\le \delta$ or $L_fV_1(\cdot)\ge -\delta$ at some point with $\|x\|>\epsilon$. At $(\delta_1,\delta_2,\omega_1,\omega_2)=(0.5,0.5,0,0)$, $V_1=0$, so if the solver returns this as a counterexample the 'verified on $D$' statement is false; if no counterexample is found, the verifier must be restricting to the sum-to-zero manifold, and that restriction should be stated as part of the certificate.

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Extended reading notes

Core claim

The central claim is that an end-to-end pipeline—a symbolic transformer that tokenizes the ODEs into prefix expressions and decodes candidate certificates, a risk-seeking policy gradient that rewards only the best candidates, SHGO-based counterexample search, and genetic-programming refinement—finds valid local Lyapunov functions for systems up to ten dimensions, and that the found expressions pass formal SMT verification. On the 2-bus lossy power-system benchmark with state space $D=\{(\delta_1,\delta_2,\omega_1,\omega_2)\in\mathbb{R}^4: |\delta_i|\le 0.75,\ |\omega_i|\le 2\}$, the paper reports $V_1=\omega_1^2+\omega_2^2+\bigl(\sin(\delta_2)-\sin(\delta_1)+\omega_2\bigr)^2$ and $V_2=\omega_1^2+\omega_2^2+\bigl(\sin(\delta_2)-\sin(\delta_1)-\omega_1\bigr)^2$ as the first analytic local Lyapunov functions for that system. The paper also recovers textbook energy functions for the pendulum and a known storage function for a 3-bus lossless power system, while a sum-of-squares baseline fails on the higher-dimensional local-stability cases.

Load-bearing premise

The claim depends on the verifier checking the Lyapunov conditions on the system's true state-space domain: the power-system examples use center-of-inertia angle coordinates whose sum is fixed at zero, while the paper states the domain as a box with no such constraint, and if the verifier silently uses the manifold the reported functions are not positive definite on the box.

Editorial extensions

If this is right

  • If the claim holds, stability certification for nonlinear control systems becomes a search problem that returns a human-readable formula rather than a black-box network, so the certificate itself can be audited and reused.
  • Final SMT verification of a found symbolic candidate takes milliseconds in the reported experiments, which would make formal guarantees cheap enough to run inside a training loop.
  • The reported success on a 10-D polynomial system and on non-polynomial dynamics suggests the approach covers systems where SOS methods need expensive recasting and where neural-Lyapunov baselines fail to converge.
  • The lossy power-system discovery, if valid, fills a specific gap: stability had been known by passivity, but no analytical Lyapunov function was available to certify it.
  • The same pipeline is proposed in the paper as extendable to other certificate functions, such as control barrier functions for safety.

Reading between the lines

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

  • One domain subtlety the paper leaves implicit: the power-system states are center-of-inertia coordinates, so the angle deviations sum to zero, while the search domain is written as a box with no such constraint; on the full box the reported $V_1$ is not positive definite, so the headline claim should be read as holding on the manifold and the verifier configuration should state that constraint exp
  • The framework's dependence on SHGO to find true minimizers suggests that as dimension grows, verification quality may degrade unless the sampling budget grows correspondingly; the paper's own ablation shows random sampling converges faster on 6-D but yields worse certificates, so a hybrid or adaptive budget is a natural next test.
  • Since dynamics constants are tokenized with rounded numerals and the symbolic library excludes general real constants, the discovered expressions are likely tailored to the exact parameter values in the benchmarks; a parameterized variant could be tested by training on a family of power-system operating points and asking whether the same template remains valid.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes an end-to-end framework that uses a symbolic transformer trained with risk-seeking policy gradient and genetic-programming expert guidance to generate candidate analytical Lyapunov functions, with SHGO-based global optimization for counterexample feedback during training and dReal for final verification. The authors report success on polynomial and non-polynomial systems up to 10 dimensions, and claim to have discovered previously unknown local Lyapunov functions for a 2-bus lossy power system. The manuscript also compares against neural Lyapunov baselines, SOS methods, and the pre-trained transformer of Alfarano et al. (2024), and provides ablations of the risk-seeking quantile, verification strategy, and GP guidance.

Significance. If the results are correct, the framework is a useful contribution: it produces concise, interpretable, analytical certificates, trains from scratch without a large pre-training dataset, scales to 10-dimensional dynamics, and ships reproducible code. The ablations are informative, and the use of an external SMT solver for final verification is a genuine safeguard against reward hacking. However, the correctness of the headline claims depends on the reported Lyapunov functions being certified on the stated state-space domains, and several of the displayed certificates appear not to satisfy the stated Lyapunov conditions on those domains. These verification/domain mismatches are load-bearing for the central contribution, so the manuscript needs substantial revision before acceptance.

major comments (3)
  1. [§5.3, Appendix G.5] The claimed 'previously unknown' Lyapunov functions for the 2-bus lossy power system are not validated as positive definite on the stated domain. The paper defines D = {(δ1,δ2,ω1,ω2) ∈ R^4 : |δi| ≤ 0.75, |ωi| ≤ 2}, but Appendix G.5 defines δi = θi − (1/N)Σθj, which for N=2 forces δ1+δ2=0. The function V1 = ω1^2 + ω2^2 + (ω2 − sin δ1 + sin δ2)^2 vanishes at δ1=δ2=c≠0, ω1=ω2=0, a point in the box D that is not in a small neighborhood of the origin. Hence V1 is not positive definite on D. If the dReal query included the implicit constraint δ1+δ2=0, that constraint is never stated; if it did not, the dReal verification cannot have succeeded as reported. The same hidden-domain issue affects the 6-D lossless power-system certificate in Appendix G.3. This must be corrected and the actual verified domain stated explicitly before the central claim can be assessed.
  2. [Table 1, Appendix F.1] The Van der Pol certificate V = x1^2 + x2^2 does not satisfy the strict Lyapunov inequality required by Proposition 3.4 on the stated domain D = {|x1|≤1, |x2|≤1}. Its Lie derivative is LfV = −2(1 − x1^2)x2^2, which vanishes on the entire line x2=0 and on the boundary x1=±1, not merely at the origin. Since the reported dReal verification uses strict inequalities V(x)>δ and LfV(x)<−δ over D\B_ε(0), points such as (0.5,0) are counterexamples to the claimed verification. The function may still certify asymptotic stability via the invariance principle, but then the verification claim and the reported SMT certification need to be revised accordingly.
  3. [Table 1, Appendix G.3] The same center-of-inertia domain issue appears in the 6-D lossless power system. Appendix G.3 introduces δi = θi − (1/N)Σθj, so the physical state space is the manifold Σδi=0, but the experiment region D is written as a box in R^6 without this constraint. On the full box, the reported function V = Σωi^2 − 0.5(Σ_{i≠j} cos(δi−δj) − 1) takes negative values, e.g. at δ1=δ2=δ3=c≠0 and ω1=ω2=ω3=0. Thus the certificate is not positive definite on the stated D. Either the verification was performed on a restricted domain that must be documented, or the table overstates the validity of the certificate.
minor comments (5)
  1. [Equation (5) vs. Appendix C] The mapping g is defined as g(x)=1/(1+x) in Equation (5), but Appendix C states 'we adopt a bounded reward function using the continuous mapping g(x)=1/x'. These are inconsistent; the intended formula is presumably 1/(1+L).
  2. [§5.3] The displayed definitions of V1 and V2 contain a typo: the intended leading term appears to be Σ_{i=1}^2 ω_i^2, but the manuscript writes 'Σ_{i=1}^2 ω1^2' for both functions.
  3. [§5.3, Appendix G.5] The paper should state explicitly whether the SHGO verification during training and the final dReal query used the box D or the manifold δ1+δ2=0; the current text makes this impossible to determine from the reported experiments.
  4. [§5.4, Table 3] The table reports '−' for SOS on the 6-D, 8-D, and 10-D polynomial systems, but the text does not specify whether this is a timeout, a solver failure, or an infeasibility result. A brief note would make the comparison reproducible.
  5. [Appendix H.2] The phrase 'Ramdom sampling' should be corrected to 'Random sampling' in the table header.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the final certificates are checked by the external dReal SMT solver, and the RL reward is only a search heuristic.

full rationale

The claimed derivation chain is: a symbolic transformer proposes analytical candidates; SHGO-based numerical verification supplies counterexamples; risk-seeking policy gradient and genetic programming refine the generator; the final candidate is formally certified with dReal (Gao et al., 2013), an external SMT solver, over D without B_epsilon(0). None of these stages defines the output in terms of itself: the reward in Eq. (5) is an empirical Lyapunov-risk heuristic used only to guide search, while the existence claim rests on the independent SMT query reported in Section 5 and Appendix G.5. The paper's self-citations (e.g., Feng et al., 2024c as a root-finding baseline and related work) are contextual comparisons, not load-bearing sources of the final certificate. The 'previously unknown' claim is a literature-availability assertion, not a parameter fit. The paper also acknowledges an optimizer limitation in Lemma C.2, but that concerns bias in the policy-gradient estimator, not the external certification step. The center-of-inertia domain ambiguity noted for the power-system examples (Appendix G.5 defines delta_i as center-of-inertia coordinates but D is written as a box in R^4) is a potential correctness/validity concern about whether the SMT query was posed on the right domain; it is not a circularity, because even if the certificate is unsupported, that would be a false or incomplete external check, not a derivation that reduces to its inputs. Thus no circular step can be exhibited, and the paper is self-contained against an external benchmark.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

No physical constants are fitted by the method itself; the fitted or hand-chosen objects are algorithm hyperparameters, test-system parameters, and verification tolerances. The load-bearing assumptions are the reliability of the SHGO and dReal solvers and the implicit use of the sum-to-zero manifold in the power-system domains. The paper introduces no new physical entities such as particles, forces, or dimensions.

free parameters (7)
  • Risk-seeking quantile alpha = 0.1
    Chosen from ablations on 3-D trig dynamics (Appendix H.1); influences training but not the formal validity of a returned certificate.
  • Candidate batch size Q = 500
    Appendix A.2; implementation choice for candidate sampling per epoch.
  • SHGO starting points = 2048
    Appendix B; selected by comparing 1024, 2048, 4096, and 8196 starting points on high-dimensional continuous functions. The adequacy of this global search is load-bearing for counterexample feedback.
  • Local sampling radius r = not specified
    Appendix B says r is small relative to ||D||2 but gives no concrete value or schedule, which affects which counterexamples are seen during training.
  • GP mutation and crossover probabilities = 0.5 each
    Appendix D; evolutionary operator rates chosen in implementation, not tied to a scientific constant.
  • Lossy power system parameters = p_i=1, m_i=2, d_i=1, u_i=omega_i, B_ij=1, G_ij=1 for i != j
    Appendix G.5; chosen by hand so the origin is the equilibrium. The claimed new lossy-system Lyapunov functions are for this specific parameter setting only.
  • SMT verification tolerances = epsilon=1e-3, delta=1e-12
    Section 5 and Appendix G.5; the strict inequalities are checked over D excluding a ball of radius epsilon, which affects which candidates pass formal verification.
assumptions (6)
  • standard math Lyapunov's asymptotic stability theorem (Proposition 3.4): if V(0)=0, V>0 in D excluding 0, and LfV<0 in D excluding 0, then the origin is asymptotically stable.
    Background theorem invoked without proof; standard in nonlinear control.
  • standard math The vector field f is Lipschitz-continuous so trajectories exist and are unique.
    Definition 3.1 assumes it; needed for the Lyapunov theorem to apply.
  • domain assumption SHGO with 2048 starting points finds global minimizers of V and -LfV reliably enough for counterexample feedback.
    Appendix B; the paper's Proposition 4.1 only says the minimizer pool stops growing once adequately sampled, not that 2048 points are adequate. Training-time verification depends on this.
  • domain assumption dReal is a trustworthy delta-complete SMT oracle for the final certificate check.
    Section 5; final verification relies on the tool's sound answers over the queried box.
  • ad hoc to paper For power-system experiments the physically relevant state space is the invariant manifold where the sum of delta_i is zero, although D is written as a box without that constraint.
    Appendix G.3 and G.5 define center-of-inertia coordinates but use box domains; the found functions are positive definite only on the manifold, so the verification domain must be the manifold for the claim to hold.
  • standard math LaSalle invariance principle: if LfV is only non-positive and no non-trivial trajectory stays in the set where LfV=0, asymptotic stability still follows.
    Used implicitly for the Van der Pol example where LfV=0 on the boundary x1=plus or minus 1, and for the quadrotor example; not stated as a separate proposition in the paper.

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Pith. "Pith review of Analytical Lyapunov Function Discovery: An RL-based Generative Approach." pith.science (2026). https://pith.science/paper/GTFBM7P4

@misc{pith2026250202014,
  author       = {Pith},
  title        = {Pith review of: Analytical Lyapunov Function Discovery: An RL-based Generative Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTFBM7P4}},
  note         = {Machine review of arXiv:2502.02014}
}
read the original abstract

Despite advances in learning-based methods, finding valid Lyapunov functions for nonlinear dynamical systems remains challenging. Current neural network approaches face two main issues: challenges in scalable verification and limited interpretability. To address these, we propose an end-to-end framework using transformers to construct analytical Lyapunov functions (local), which simplifies formal verification, enhances interpretability, and provides valuable insights for control engineers. Our framework consists of a transformer-based trainer that generates candidate Lyapunov functions and a falsifier that verifies candidate expressions and refines the model via risk-seeking policy gradient. Unlike Alfarano et al. (2024), which utilizes pre-training and seeks global Lyapunov functions for low-dimensional systems, our model is trained from scratch via reinforcement learning (RL) and succeeds in finding local Lyapunov functions for high-dimensional and non-polynomial systems. Given the analytical nature of the candidates, we employ efficient optimization methods for falsification during training and formal verification tools for the final verification. We demonstrate the efficiency of our approach on a range of nonlinear dynamical systems with up to ten dimensions and show that it can discover Lyapunov functions not previously identified in the control literature. Full implementation is available on \href{https://github.com/JieFeng-cse/Analytical-Lyapunov-Function-Discovery}{Github}

Figures

Figures reproduced from arXiv: 2502.02014 by the authors.

Figure 1
Figure 1. We present the statistics of runtime for single SMT ver￾ification (precision e −6 , tolerance error e −1 ) on the 6-D system (Appendix F.4). Each method is validated once per epoch. The red line represents the median. The mean verification time is averaged over one complete training, and the value is displayed in bold. Neural Network in FOSSIL 2.0 Edwards et al. (2024) has a 10-D hidden layer, and Augmented Neural L… view at source ↗
Figure 2
Figure 2. Framework overview: The transformer takes embeddings of the dynamical system model as input and generates candidate Lyapunov functions in an autoregressive manner. Hierarchical information is deployed to enhance the model input. For example, when generating the last token x2, its parent is +, and its sibling is x1. The output is the pre-order traversal of the expression’s binary tree. Candidates are verified using a… view at source ↗
Figure 3
Figure 3. We visualize the dynamics tokenization process of the simple pendulum system. Each analytical formula in the ODE representa￾tion of input dynamics is first converted into an expression tree and then represented by the pre-order traversal in symbolic tokens and constant coefficients. Computation Resources: All experiments in this work were performed on a workstation with an AMD Ryzen Threadripper 2920X 12-Core Proces… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: This plot visualizes our proposed verification process on a sampled candidate. V˜ϕ = (x1 + x2) 2 + x2 is a sampled candidate during the training for Van Der Pol Oscillator. Using Simplicial Homology Global Optimization, we first identify the minimizer of the Lyapunov f…
Figure 5
Figure 5. Figure 5: Reward Trajectory Comparison for Ablation Study I: Test proposed framework (without GP component) on 3-D Trig dynamics with three different choices of α for 100 epochs. The figure on the left visualizes the highest reward in the batch of samples obtained in each epoch,…
Figure 6
Figure 6. Figure 6: Best Reward Trajectory for Ablation Study II: Visual￾ize the best reward trajectory of our framework with SHGO, root finding, and random sampling schemes in Ablation Study II on 6-D polynomial dynamics. The SMT-verification-based trajectory is not included as it timeou…
Figure 8
Figure 8. Figure 8: The figure on the left visualizes the overall best reward (best candidate among V˜ϕ ∪ V˜gp) trajectory. The figure on the left visualizes the best reward trajectory for transformer candidates (best candidate among V˜ϕ only, excludes GP-generated solutions). These two p…

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