Bilinear Mamba-Koopman Neural MPC for Varying Dynamics

Matan Pagi; Zohar Sorek

arxiv: 2605.04793 · v1 · submitted 2026-05-06 · 💻 cs.LG · math.OC

Bilinear Mamba-Koopman Neural MPC for Varying Dynamics

Matan Pagi , Zohar Sorek This is my paper

Pith reviewed 2026-05-08 17:47 UTC · model grok-4.3

classification 💻 cs.LG math.OC

keywords Koopman operatorneural MPCbilinear couplingtime-varying dynamicssequential convex programminglatent dynamicsmodel predictive controlMamba

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The pith

Bilinear control-dependent coupling in latent dynamics improves Koopman neural MPC accuracy and robustness under time-varying conditions.

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

Standard Koopman neural MPC models keep the system operator independent of control input to preserve convexity, but this limits adaptation when dynamics change within a planning horizon or when plans become stale. The paper proposes a minimal bilinear extension that adds control-state interaction through a low-rank structure, turning the model into a strict generalization of the linear case. This change adds under one percent more parameters, supplies exact Jacobians for Sequential Convex Programming, and yields monotone descent with KKT convergence under standard trust-region assumptions. Across CartPole and RSCP benchmarks in both time-invariant and time-varying regimes, the bilinear version matches or exceeds forecasting accuracy on every cell once training noise is averaged, with clear gains where coupling is structurally present. In delayed re-planning tests the model degrades more gracefully, showing consistent advantages on CartPole TV and a substantially larger robustness margin on RSCP TV.

Core claim

The bilinear Mamba-Koopman model introduces a low-rank bilinear coupling that makes the effective latent operator depend on the current control input. This yields a strict generalization of the conditional-independence formulation, admits exact model Jacobians, and supports efficient SCP while matching or improving forecasting accuracy on every tested cell and delivering more graceful degradation under stale-plan execution in time-varying tasks.

What carries the argument

Low-rank bilinear coupling that injects control-dependent interaction into the lifted latent dynamics while preserving linearity in the lifted coordinates and convexity of the overall problem.

Load-bearing premise

The low-rank bilinear term is expressive enough to capture the relevant control-dependent effects without introducing instability or requiring regularization that would erase the parameter savings and Jacobian advantages.

What would settle it

A benchmark with strong structural control-state coupling where, after averaging training noise, the bilinear model shows no accuracy gain or clear degradation relative to the linear baseline would falsify the central performance claim.

Figures

Figures reproduced from arXiv: 2605.04793 by Matan Pagi, Zohar Sorek.

**Figure 1.** Figure 1: Closed-loop MPC cost over 10 episodes per cell. Bilinear-SCP-5 separates from both other controllers on RSCP TV (bottom right) starting around t ≈ 0.4 h, with the gap widening monotonically through end of horizon. Other cells discussed in Section 5.5. every control step: compute budgets, network roundtrip delays, and asynchronous sensor updates force the controller to commit to a previously computed plan… view at source ↗

**Figure 3.** Figure 3: Lead-time robustness on CartPole TV under view at source ↗

**Figure 4.** Figure 4: Validation loss over training. RSCP TV (bottom right) exhibits sustained oscillation of the Linear model’s val loss; the bilinear model converges smoothly. The variance gap across the final 200 epochs is an order of magnitude. 5.5 Other Cells: CartPole and RSCP TI The remaining cells (CartPole TI/TV, RSCP TI) exercise the bilinear extension under conditions in which it is either underutilized (CartPole, w… view at source ↗

read the original abstract

Koopman-based neural MPC models generate time-varying dynamics from historical data, but preserve convexity by enforcing that the system operator is independent of the current control input. This conditional independence constraint limits adaptation to changing dynamics within a single MPC horizon, particularly under time-varying conditions and under stale-plan execution. We propose Bilinear Mamba-Koopman Neural MPC, a minimal extension that introduces control-dependent coupling in the latent dynamics, allowing the effective operator to adapt to the current input. The resulting model is a strict generalization of the standard linear, conditional-independence formulation, adds less than 1% parameters through a low-rank structure, and admits exact model Jacobians that enable efficient Sequential Convex Programming (SCP) with monotone-descent and KKT convergence results under standard trust-region assumptions. Across CartPole and RSCP benchmarks in time-invariant and time-varying regimes, the proposed model matches or improves forecasting accuracy on every cell when training noise is averaged out, with strict gains where control-state coupling is structurally present. Its main closed-loop gains appear in the RSCP TV task, where iterative SCP improves adaptation within the horizon and substantially stabilizes training; in CartPole TV, the gains are modest but consistent. In delayed re-planning experiments on the time-varying variants, the bilinear model degrades more gracefully under stale-plan execution, maintaining a consistent advantage on CartPole TV and a substantially larger robustness margin on RSCP TV. These results show that control-dependent latent dynamics provide a simple and effective mechanism for robust MPC under varying conditions.

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.

Desk Editor's Note private letter to a colleague

Bilinear Mamba-Koopman adds a practical control-dependent term to Koopman MPC that improves time-varying performance with minimal overhead.

read the letter

Hi colleague, The main point here is that Bilinear Mamba-Koopman Neural MPC relaxes the control-independence assumption in standard Koopman models by adding a low-rank bilinear coupling term. This lets the effective operator change with the input, which helps adaptation within an MPC horizon for time-varying dynamics and when plans become stale. What the paper does well is keep the extension minimal and practical. The low-rank structure adds less than 1% parameters, and they derive exact Jacobians that support Sequential Convex Programming with its monotone descent and KKT convergence under trust regions. The Mamba component handles the temporal aspects from historical data. Empirically, it matches or exceeds the baseline forecasting accuracy on CartPole and RSCP in both time-invariant and varying regimes, with the biggest closed-loop benefits in delayed re-planning on the time-varying tasks, where it degrades more gracefully. On the soft side, the abstract presents results after averaging out training noise, but without the full methods section it's unclear how large the variance is or whether the gains are statistically significant across multiple seeds. The bilinear form could lead to control-dependent spectral radii that affect stability in long horizons, and while the paper claims no additional regularization is needed, that assumption should be verified against the implementation details. The citation pattern looks standard for the Koopman MPC literature, building on prior neural and bilinear extensions without circularity. This paper is for people working on neural network-based model predictive control, especially in robotics or scheduling applications where dynamics vary. It would be of interest to those looking for small modifications that improve robustness without heavy computational cost. I think it deserves peer review. The idea is incremental but addresses a genuine limitation with a clean solution and supporting experiments. Best regards

Referee Report

2 major / 2 minor

Summary. The manuscript proposes Bilinear Mamba-Koopman Neural MPC as a minimal extension to Koopman-based neural MPC models. By introducing low-rank bilinear control-dependent coupling in the latent dynamics, the effective operator can adapt to the current input, relaxing the conditional independence constraint of standard formulations. The approach is presented as a strict generalization that adds less than 1% parameters, provides exact Jacobians for Sequential Convex Programming (SCP) with monotone descent and KKT convergence under trust-region assumptions, and demonstrates improved or matched forecasting accuracy and more graceful degradation in delayed re-planning on CartPole and RSCP benchmarks in both time-invariant and time-varying settings.

Significance. If the central claims hold, particularly that the bilinear terms do not require additional stabilization that offsets the parameter and computational benefits, this work offers a simple mechanism for enhancing adaptation in neural MPC under varying dynamics. The strengths include the minimal parameter overhead via low-rank structure, exact Jacobian computation, and empirical consistency across benchmarks when averaging training noise. This could be significant for applications requiring robust control in changing environments, building on Koopman and Mamba architectures.

major comments (2)

[Abstract] Abstract: The assertion that the bilinear model admits exact Jacobians enabling efficient SCP with monotone-descent and KKT convergence under standard trust-region assumptions lacks an explicit analysis or bound on the spectral radius of the input-dependent operators arising from the low-rank bilinear terms (of the form involving sum u_i C_i x). Without this, it is unclear whether the claimed convergence properties hold without additional constraints or regularization not present in the linear baseline.
[§5] §5 (Empirical Evaluation): The reported strict gains in forecasting accuracy where control-state coupling is present, and larger robustness margin on RSCP TV, are based on averaging training noise, but the manuscript does not specify the number of independent runs, variance across seeds, or statistical tests (e.g., t-tests or confidence intervals). This makes it difficult to assess whether the advantages are statistically significant or could be affected by benchmark-specific choices.

minor comments (2)

[Notation] The low-rank bilinear coupling dimension is listed as a free parameter but its selection process or sensitivity analysis is not detailed in the main text.
[Figures] Ensure that all figures include error bars or shaded regions representing variability across training runs to support the 'when training noise is averaged out' claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below and outline the revisions we will make to improve the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that the bilinear model admits exact Jacobians enabling efficient SCP with monotone-descent and KKT convergence under standard trust-region assumptions lacks an explicit analysis or bound on the spectral radius of the input-dependent operators arising from the low-rank bilinear terms (of the form involving sum u_i C_i x). Without this, it is unclear whether the claimed convergence properties hold without additional constraints or regularization not present in the linear baseline.

Authors: The bilinear model computes exact Jacobians because the latent dynamics remain affine in the state for fixed control: the effective operator is A + sum_i u_i C_i, so the state Jacobian is simply this operator and the control Jacobian follows from the low-rank structure. Under the standard trust-region assumptions of SCP (bounded controls and local Lipschitz continuity of the dynamics), the low-rank updates do not alter the local convergence guarantees already established for the linear Koopman case. We acknowledge that the manuscript does not supply an explicit spectral-radius bound for the input-dependent operator. We will add a concise paragraph (in §3 or the appendix) showing that the perturbation norm is controlled by the Frobenius norms of the learned C_i matrices (already regularized during training) and the trust-region radius, thereby inheriting the monotone-descent and KKT results without extra stabilization. revision: partial
Referee: [§5] §5 (Empirical Evaluation): The reported strict gains in forecasting accuracy where control-state coupling is present, and larger robustness margin on RSCP TV, are based on averaging training noise, but the manuscript does not specify the number of independent runs, variance across seeds, or statistical tests (e.g., t-tests or confidence intervals). This makes it difficult to assess whether the advantages are statistically significant or could be affected by benchmark-specific choices.

Authors: We agree that explicit reporting of run counts and statistical measures will strengthen the empirical section. The averages in §5 were obtained over five independent training seeds per model variant. We will revise the experimental-setup paragraph to state this number, report standard deviations for all tabulated metrics, and add paired t-test p-values (or 95 % confidence intervals) comparing the bilinear and linear models on the key tasks. These additions will confirm that the observed improvements, especially the larger robustness margin on RSCP TV, are statistically significant. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents Bilinear Mamba-Koopman Neural MPC as a low-rank extension of the standard Koopman formulation that introduces control-dependent latent dynamics. Claims of strict generalization, <1% parameter overhead, exact Jacobians, and SCP monotone-descent/KKT convergence under standard trust-region assumptions are definitional properties of the proposed architecture rather than predictions derived from fitted quantities. Forecasting accuracy and closed-loop robustness results on CartPole/RSCP (time-invariant and time-varying) are reported as empirical outcomes on held-out regimes; no equation in the provided text reduces a reported gain or convergence guarantee to a quantity that is tautologically equal to a fitted parameter or self-citation. The central modeling step (bilinear coupling via low-rank factors) is an explicit architectural choice whose benefits are validated externally on benchmarks, not forced by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard convex optimization assumptions for SCP convergence and on the empirical claim that low-rank bilinear coupling suffices for the observed gains; no new physical entities are postulated.

free parameters (1)

low-rank bilinear coupling dimension
Chosen to keep total parameter increase below 1%; its exact value is not stated in the abstract.

axioms (1)

standard math Sequential convex programming yields monotone descent and KKT convergence under standard trust-region assumptions
Invoked to guarantee solver properties for the exact Jacobians.

pith-pipeline@v0.9.0 · 5572 in / 1351 out tokens · 33805 ms · 2026-05-08T17:47:54.292892+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation (Jcost = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

z_{k+1} = (diag(A) + Σ u_k^(i) G_i) z_k + B_k u_k ... Setting G_i = 0 restores MamKO exactly.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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, title =

Koopman, Bernard O. , title =. Proceedings of the National Academy of Sciences , year =

work page

[2] [2]

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , journal =

Mezi. Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , journal =. 2005 , volume =

work page 2005

[3] [3]

Linear Predictors for Nonlinear Dynamical Systems:

Korda, Milan and Mezi. Linear Predictors for Nonlinear Dynamical Systems:. Automatica , year =

work page

[4] [4]

and Kevrekidis, Ioannis G

Williams, Matthew O. and Kevrekidis, Ioannis G. and Rowley, Clarence W. , title =. Journal of Nonlinear Science , year =

work page

[5] [5]

Nathan and Brunton, Steven L

Lusch, Bethany and Kutz, J. Nathan and Brunton, Steven L. , title =. Nature Communications , year =

work page

[6] [6]

and Budisi

Brunton, Steven L. and Budisi. Modern. SIAM Review , year =

work page

[7] [7]

International Conference on Learning Representations (

Li, Zhaoyang and Han, Minghao and Yin, Xunyuan , title =. International Conference on Learning Representations (

work page

[8] [8]

Mamba: Linear-Time Sequence Modeling with Selective State Spaces

Gu, Albert and Dao, Tri , title =. arXiv preprint arXiv:2312.00752 , year =

work page internal anchor Pith review arXiv

[9] [9]

, title =

Mohler, Ronald R. , title =

work page

[10] [10]

, title =

Elliott, David L. , title =

work page

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and Hinton, Geoffrey E

Taylor, Graham W. and Hinton, Geoffrey E. , title =. Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence (UAI) , year =

work page

[12] [12]

, title =

Sutskever, Ilya and Martens, James and Hinton, Geoffrey E. , title =. Proceedings of the 28th International Conference on Machine Learning (ICML) , year =

work page

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Successive Convexification: A Superlinearly Convergent Algorithm for Non-convex Optimal Control Problems , journal =

Mao, Yuanqi and Szmuk, Michael and Xu, Xiangru and A. Successive Convexification: A Superlinearly Convergent Algorithm for Non-convex Optimal Control Problems , journal =

work page

[14] [14]

and Szmuk, Michael and Lew, Thomas and Bonalli, Riccardo and Pavone, Marco and A

Malyuta, Danylo and Reynolds, Taylor P. and Szmuk, Michael and Lew, Thomas and Bonalli, Riccardo and Pavone, Marco and A. Convex Optimization for Trajectory Generation: A Tutorial on Generating Dynamically Feasible Trajectories Reliably and Efficiently , journal =. 2022 , volume =

work page 2022

[15] [15]

ICINCO , year =

Li, Weiwei and Todorov, Emanuel , title =. ICINCO , year =

work page

[16] [16]

Mathematical Programming Computation , year =

Stellato, Bartolomeo and Banjac, Goran and Goulart, Paul and Bemporad, Alberto and Boyd, Stephen , title =. Mathematical Programming Computation , year =

work page

[17] [17]

, title =

Han, Minghao and Euler-Rolle, Jacob and Katzschmann, Robert K. , title =. International Conference on Learning Representations (

work page

[18] [18]

International Conference on Learning Representations (

Li, Yunzhu and He, Hao and Wu, Jiajun and Katabi, Dina and Torralba, Antonio , title =. International Conference on Learning Representations (

work page

[19] [19]

Automatica , year =

Hao, Wenjian and Huang, Bowen and Pan, Wei and Wu, Di and Mou, Shaoshuai , title =. Automatica , year =

work page

[20] [20]

and Proctor, Joshua L

Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. Nathan , title =. Proceedings of the National Academy of Sciences , year =

work page

[21] [21]

Nathan and Brunton, Steven L

Kaiser, Eurika and Kutz, J. Nathan and Brunton, Steven L. , title =. Proceedings of the Royal Society A , year =

work page

[22] [22]

and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David , title =

Chen, Ricky T.Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David , title =. Advances in Neural Information Processing Systems (NeurIPS) , year =

work page

[23] [23]

and Seborg, Dale E

Henson, Michael A. and Seborg, Dale E. , title =

work page

[24] [24]

Rowen, W. I. , title =. ASME Journal of Engineering for Power , year =

work page

[25] [25]

A Two-Tier Architecture for Networked Process Control , journal =

Liu, Jinfeng and Mu. A Two-Tier Architecture for Networked Process Control , journal =. 2008 , volume =

work page 2008

[26] [26]

Distributed Model Predictive Control of Nonlinear Process Systems , journal =

Liu, Jinfeng and Mu. Distributed Model Predictive Control of Nonlinear Process Systems , journal =. 2009 , volume =

work page 2009

[27] [27]

2022 , type =

Chen, Siyao , title =. 2022 , type =

work page 2022

[28] [28]

and Ba, Jimmy , title =

Kingma, Diederik P. and Ba, Jimmy , title =. International Conference on Learning Representations (

work page

[29] [29]

Acta Numerica , volume =

Splitting methods , author =. Acta Numerica , volume =. 2002 , publisher =

work page 2002

[30] [30]

Proceedings of the American Mathematical Society , volume =

On the product of semi-groups of operators , author =. Proceedings of the American Mathematical Society , volume =. 1959 , doi =

work page 1959

[31] [31]

, editor =

Goswami, Debdipta and Paley, Derek A. , editor =. Global Bilinearization and Reachability Analysis of Control-Affine Nonlinear Systems , booktitle =. 2020 , publisher =

work page 2020