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REVIEW 5 major objections 5 minor 28 references

Data-driven optimal control of unknown nonlinear dynamical systems using the Koopman operator

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A data-driven controller trained on sampled trajectories can match the value function, control law, and closed-loop cost of a controller designed with full knowledge of the underlying nonlinear dynamics.

desk verdict Solid empirical paper with an overreaching theorem: the random-feature GHJB fit is not covered by the convergence proof, but the identification improvement is real and the experiments are convincing. read the letter →

arxiv 2412.01085 v1 pith:4VOPFHFN submitted 2024-12-02 eess.SY cs.SY

Zhexuan Zeng , Ruikun Zhou , Yiming Meng , Jun Liu This is my paper · ORCID
classification eess.SYcs.SY
keywords KoopmanoperatorsystemidentificationoptimalcontrolpolicyiterationHamilton-Jacobi-BellmanequationYosidaapproximationrandomfeaturenetworkshigh-dimensionalnonlinearsystems
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

This paper asks whether a near-optimal stabilizing controller for an unknown nonlinear system can be learned from sampled trajectories alone, with the same guarantees as a model-based design. It claims yes: identify the drift $f$ and actuation $g$ from data through a modified Koopman-generator estimate, then run policy iteration where each step solves the generalized Hamilton-Jacobi-Bellman equation by a random-feature least-squares fit. The authors prove that, at each iteration, the learned value function and control law converge uniformly to those of the true system as identification resolution and function-approximation capacity grow. In experiments on systems with 2 to 9 states, the accumulated cost of the learned control matches a controller trained with full model knowledge to within $10^{-5}$ to $10^{-3}$.

What carries the argument

The load-bearing object is the Yosida approximation of the Koopman generator, $L_\lambda \varphi = \lambda^2 \int_0^\infty e^{-\lambda t} U^t \varphi \, dt - \lambda \varphi$, truncated to a finite horizon and evaluated on sampled data by quadrature, then fitted by least squares to give a matrix representation of the generator. Using observables that are polynomial in $x$ and linear in $u$ lets the identified generator directly recover the control-affine terms $f(x) + g(x)u$. The value-function step is carried by a random-feature network $\hat{V}(x) = \beta^T \sigma(Wx + b)$ with random $W$ and $b$, which turns each GHJB policy-evaluation equation into a linear least-squares problem for $\beta$ and makes the iteration scalable to state dimension 9.

What would settle it

Run the pipeline on a control-affine system with a known closed-form GHJB solution, record the residual of the random-feature least-squares GHJB solve at each policy-iteration step, and compare the realized value-function error to the error Theorem 3 would bound from the identified dynamics alone; if the residual is not small, the computed controller can violate the claimed guarantee. Alternatively, run the 3D quadrotor test on a domain substantially larger than $[-0.1, 0.1]^9$ and check whether the $10^{-5}$ to $10^{-3}$ accumulated-cost error persists.

Watch

Extended reading notes

Core claim

The central claim is that a fully data-driven procedure can reproduce the result of exact policy iteration for a control-affine nonlinear system. The procedure first samples trajectories of the unknown dynamics, uses a Yosida-type approximation to estimate the Koopman generator on an augmented state-input space, and recovers estimates of $f(x)$ and $g(x)$. It then solves the GHJB equation at each policy-evaluation step by fitting a random-feature neural network through linear least squares, which keeps the iteration tractable. The paper's Theorem 3 states that if the identified vector field and the initial policy are close enough to the true ones, the value function and policy obtained at every iteration are uniformly close to the exact ones; Theorem 2 provides the convergence of the identified vector field as $\lambda \to \infty$, $T_{\max} \to \infty$, and basis size $N \to \infty$. The numerical section demonstrates stabilizing control and cost errors from $10^{-5}$ to $10^{-3}$ on inverted pendulum, cartpole, 2D quadrotor, and 3D quadrotor examples.

Load-bearing premise

The argument assumes that dense enough sampling makes the identified vector field close to the truncated Yosida generator, and that the random-feature least-squares fit is close enough to the exact GHJB solution for Theorem 3 to apply; the paper does not bound the second gap.

Editorial extensions

If this is right

  • A controller can be synthesized for a 9-state, 4-input nonlinear plant without a model, and still stabilize the true system in simulation.
  • Each policy-iteration step inherits a uniform convergence guarantee: if the identified dynamics are close to the true dynamics, the value function and policy are close to the exact ones at that step.
  • The closed-loop accumulated cost of the learned controller tracks the cost of a full-model controller to within $10^{-5}$ to $10^{-3}$ on the tested domains.
  • The improved identification accuracy (one to two orders of magnitude lower $f$ and $g$ error than the log-based and lifted-linear baselines) is what makes the HJB solution usable in higher dimensions.

Reading between the lines

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

  • A natural next step would be to turn the least-squares residual of the random-feature GHJB solve into a data-dependent certificate, since the paper's convergence proof assumes the exact GHJB solution is available.
  • If the same identification step is combined with a different PDE solver (for example a physics-informed network), the policy-iteration guarantee would likely transfer, because the theorem only requires closeness of the value function and its gradient.
  • The current polynomial observables are linear in each control input; extending to non-affine systems would require observables with higher powers of $u$, which would change the recovery step and is not covered by the analysis.
  • The cost-error range $10^{-5}$ to $10^{-3}$ is reported on small compact domains; testing the method on larger or less regular operating regions would reveal how quickly the guarantee degrades with domain size.
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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

5 major / 5 minor

Summary. The paper proposes a data-driven optimal control framework for unknown control-affine nonlinear systems. The method first identifies the dynamics by estimating the Koopman generator via a Yosida approximation with polynomial observables that include control-affine coupling terms, then solves the optimal control problem by policy iteration in which the Generalized Hamilton-Jacobi-Bellman (GHJB) equation is approximately solved with random-feature neural networks. The authors claim that the identified vector field converges to the true vector field and that the value function and control policy converge at each iteration to those of the true system. Numerical experiments on inverted pendulum, cartpole, 2D quadrotor, and 3D quadrotor systems report low identification errors and accumulated-cost errors between 10^-5 and 10^-3 relative to a model-based oracle. The central theoretical result, Theorem 3, is meant to certify the convergence of the value function and policy, but the proof only covers exact solutions of the GHJB equation, not the random-feature least-squares solution actually computed by the algorithm.

Significance. If the theoretical gaps were closed, the paper would be a valuable contribution. The empirical results are strong: the resolvent-based control-affine identification improves identification errors by one to two orders of magnitude over the logarithm-based and lifted-linear baselines, and the learned controllers stabilize true 9-dimensional systems with cost errors on the order of 1e-5 to 1e-3. The paper also explicitly avoids the operator-logarithm step, which is a technically useful idea. However, the claimed convergence guarantee does not currently cover the actual algorithm, and the proof of Theorem 3 contains sign and regularity issues. These are load-bearing problems for the paper's main theoretical claim, so the manuscript needs substantial revision before it can be accepted.

major comments (5)
  1. [Section 4.4 and Theorem 3] Theorem 3 in Section 5, proved in Appendix A, applies only to the exact solutions V_h^{(i)} of the GHJB equations (19)-(20). The algorithm in Section 4.4, however, does not solve these equations; it fits a random-feature ansatz V(x) = beta^T sigma(Wx+b) by linear least squares. The paper provides no residual bound, no representability condition, and no consistency estimate relating the least-squares solution to the exact GHJB solution. Therefore the abstract's claim that 'the learned value function and control laws are proven to converge' is not justified for the quantities actually computed in the experiments. This is the main gap in the theoretical certificate.
  2. [Appendix A, Eqs. (23)-(24)] In the proof of Theorem 3, L_h^{(i)} is defined as -Q(x) - kappa_{h}^{(i-1)}(x)^T R kappa_{h}^{(i-1)}(x), which is negative definite. The tail integrals in (23)-(24) are therefore non-positive, so the inequalities 0 <= integral_{T1}^infinity L_h^{(i)} dt <= theta/2 cannot hold as written. The proof of the value-function bound |V_h^{(i)} - V^{(i)}| <= theta is thus not established. The sign convention in the GHJB equations and the subsequent integral representation must be reconciled, and absolute values should be used for the tail estimates.
  3. [Appendix A, Eqs. (50) and (52)] The proof of the control-law convergence uses the bound ||D L^{(i)} - D L_h^{(i)}||_infinity <= C_L ||L^{(i)} - L_h^{(i)}||_infinity in Eq. (50) and assumes in Eq. (52) that D L_h^{(i)} is Lipschitz. Neither statement follows from the theorem's assumption that L^{(i)}, L_h^{(i)} in C^1(Omega). These estimates require at least C^2 regularity or Lipschitz gradients. The assumptions of Theorem 3 must be strengthened or the proof revised accordingly.
  4. [Section 5, Assumption 1] Assumption 1 simply postulates that sufficiently dense sampling makes the identified vector field uniformly close to the truncated Yosida vector field. No sample-complexity bound, noise model, or relation between the number of samples M, the truncation parameters lambda and Tmax, and the finite-dimensional subspace N is given. Table 3 reports average identification errors for the specific test systems, but this does not verify the uniform closeness over the whole compact set M required for the subsequent conclusion (18). This assumption is load-bearing for the identification part of the theory and should either be proved for the proposed least-squares estimator or explicitly tested and reported.
  5. [Section 4.4 and Section 6] The paper does not explain how an initial admissible (stabilizing) policy kappa_h^{(0)} is obtained from data, even though Theorem 3 assumes ||kappa_h^{(0)} - kappa^{(0)}||_infinity < delta and policy iteration requires a stabilizing initial controller. Without a data-driven procedure for computing such an initial policy, the practical applicability of the method is not fully demonstrated, and this requirement should be discussed explicitly.
minor comments (5)
  1. [Section 6.1] Several parameter values needed for reproducibility are missing: the Yosida parameter lambda, the truncation horizon Tmax, the numerical quadrature scheme for (13), and the activation function and distribution for the random features W and b in Section 4.4 are not reported in Tables 1-2 or in the text.
  2. [Section 6.2, Figure 1] The left panels of Figure 1 plot 'Mean error' with scale factors (1e-4, 1e-3, 1e-5) but do not state whether the displayed quantity is the absolute cost difference |C_hat(t)-C(t)| or a relative error, and no error bars are provided.
  3. [Section 6.1] There are typos in the table and text: '2D quatroter' and '2D quadroter' should be '2D quadrotor', and 'Hidden unites' should be 'Hidden units'.
  4. [Section 5] The sentence 'The proof is provided in the arXiv version (Zeng et al., 2024c)' is unnecessary because the appendix contains a full proof of Theorem 3; this reference to a preprint of the same paper should be removed or corrected.
  5. [Section 4.3, Eq. (15)] The notation [L_hat_N]_{k,i} uses the index i, which also denotes the policy-iteration step; this dual use is confusing and should be disambiguated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; the known approximation gap is a correctness concern, not a definitional reduction.

full rationale

The claimed convergence results relate quantities that are defined independently rather than in terms of one another. The identified vector field is obtained from the Yosida approximation of the Koopman generator, the value functions are defined as integral costs, and the control laws are defined through the Hamiltonians of the identified model. Theorem 3 and its Appendix A proof concern the exact GHJB solutions of the identified and true systems, and the proof is a standard continuity argument using Gronwall-type estimates; it does not assume the computed random-feature least-squares coefficient beta equals the exact GHJB solution. That leaves a real representability gap between the algorithm of Section 4.4 and the theorem, but the gap is a missing residual bound rather than a circular step, since the theorem's hypotheses do not include the conclusion. The reliance on prior work (Meng et al. 2024b for Yosida convergence and error bounds, Liu et al. 2023 for derivative formulas) is self-citation, but those are stated mathematical results with hypotheses that do not themselves assert the present paper's convergence claims, so they are independent support rather than load-bearing circularity. The reported cost errors are computed by forward simulation of the true system under the learned controller, giving genuine comparisons against a model-based oracle rather than fitted inputs renamed as predictions. No step reduces to its own inputs by construction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The free parameters are mostly experimental hyperparameters that are underreported. The main axiomatic burden is Assumption 1 and the unverified representability of the GHJB solution in the random-feature space. The proof of Theorem 3 implicitly assumes C^2 smoothness for L_h, which is not stated in the theorem.

free parameters (6)
  • lambda (Yosida parameter) = not reported
    The generator approximation L_{lambda,Tmax} depends on lambda; convergence requires lambda to infinity, but the numerical values used in the experiments are not given.
  • Tmax (truncation horizon) = 1 (implied by data horizon t in [0,1])
    The truncated integral in (13) over [0,Tmax] is a finite approximation; the paper reports the data collection horizon but does not state Tmax explicitly.
  • Polynomial order pmax or psum = 5 (2D), 3 (others)
    The observable space size is chosen per system and directly affects identification accuracy; no sensitivity analysis is provided.
  • Number of random features s = 200 (2D), 3200 (others)
    The hidden units in the random-feature network are chosen per system with no ablation or justification.
  • Cost weighting matrices Q and R = not reported
    The optimal control problem is defined by L(x,u) = Q(x) + ||u||^2_R, but the specific Q and R used in the experiments are never stated.
  • Random feature weights W and b distribution = not reported
    The random-feature network draws W and b randomly, but the distribution is not specified, which is necessary for replication.
assumptions (4)
  • domain assumption Sampled initial conditions for the extended system (11) are sufficiently dense in M so that the identified vector field \hat F from (15) satisfies ||F_{lambda,Tmax,N} - \hat F||_infty <= theta (Assumption 1).
    This is load-bearing for identification accuracy and is never verified in the experiments. It appears in Section 5.
  • domain assumption The GHJB solution V^{(i)} and V_h^{(i)} are C^1 and the value functions are finite Lyapunov functions on Omega.
    Required for Theorem 3 and for the integral representations of the value functions in the appendix.
  • ad hoc to paper L_h^{(i)} is C^1 on Omega, and in the proof of Theorem 3 the derivative DL_h is assumed Lipschitz, which requires C^2 smoothness.
    The proof uses a Lipschitz constant LipDL_h for DL_h (appendix, around Eq. 52) without stating this C^2 assumption in the theorem statement.
  • standard math The system is control-affine with f in C^1, g smooth, f(0)=0, and 0 is an equilibrium.
    This is the standard problem formulation in Section 2, and is inherited from the control literature.

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Cite this review

Pith. "Pith review of Data-driven optimal control of unknown nonlinear dynamical systems using the Koopman operator." pith.science (2026). https://pith.science/paper/4VOPFHFN

@misc{pith2026241201085,
  author       = {Pith},
  title        = {Pith review of: Data-driven optimal control of unknown nonlinear dynamical systems using the Koopman operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VOPFHFN}},
  note         = {Machine review of arXiv:2412.01085}
}
abstract

Nonlinear optimal control is vital for numerous applications but remains challenging for unknown systems due to the difficulties in accurately modelling dynamics and handling computational demands, particularly in high-dimensional settings. This work develops a theoretically certifiable framework that integrates a modified Koopman operator approach with model-based reinforcement learning to address these challenges. By relaxing the requirements on observable functions, our method incorporates nonlinear terms involving both states and control inputs, significantly enhancing system identification accuracy. Moreover, by leveraging the power of neural networks to solve partial differential equations (PDEs), our approach is able to achieving stabilizing control for high-dimensional dynamical systems, up to 9-dimensional. The learned value function and control laws are proven to converge to those of the true system at each iteration. Additionally, the accumulated cost of the learned control closely approximates that of the true system, with errors ranging from $10^{-5}$ to $10^{-3}$.

Figures

Figures reproduced from arXiv: 2412.01085 by the authors.

Figure 1
Figure 1. The error between accumulated costs computed by the control learned from identified system and the true system (left). 50 trajectories of all states driven by the control learned from the identified system (right). 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

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