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arxiv: 2607.01203 · v1 · pith:6SRVHE4Dnew · submitted 2026-07-01 · 📡 eess.SY · cs.AI· cs.LG· cs.RO· cs.SY· math.OC

GPU-Parallel Linearization Error Bounds for Real-Time Robust Optimal Control of Nonlinear and Neural Network Dynamics

Jeffrey Fang , Keyi Shen , Anutam Srinivasan , Glen Chou This is my paper

Pith reviewed 2026-07-02 07:09 UTC · model grok-4.3

classification 📡 eess.SY cs.AIcs.LGcs.ROcs.SYmath.OC
keywords linearization error boundsrobust optimal controlGPU accelerationneural network dynamicssystem-level synthesisreachable setsreal-time controluncertain nonlinear systems
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The pith

GPU-parallel linearization error bounds enable real-time robust optimal control for nonlinear and neural network dynamics.

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

The paper develops methods to compute tight, differentiable linearization error bounds on GPUs for linear time-varying approximations of nonlinear and neural network systems. These bounds are integrated into a robust control solver to optimize feedback policies online while ensuring formal verification of constraint satisfaction through reachable tubes. This approach addresses the challenge of balancing tractability, tightness, and real-time performance in uncertain dynamics up to 168 dimensions, achieving rates up to 67 Hz. A sympathetic reader would care because it makes robust control feasible for complex systems without excessive conservativeness or loss of guarantees.

Core claim

The central claim is that path-based Hessian bounds for analytic dynamics and verifier-generated affine relaxations with Jacobian corrections for NN dynamics produce sound and tight LEBs that, when used in an adapted GPU-parallel system-level synthesis solver handling right-invertible disturbances and non-zero-centered sets, allow online computation of robust policies with tight formally verified reachable tubes.

What carries the argument

GPU-parallel linearization error bounds (LEBs) within the GPUSLS-LEO framework, combining path-based Hessian bounds or NN verifier relaxations with an adapted SLS solver for zonotopic propagation.

If this is right

  • Robust feedback policies accounting for linearization error can be optimized online at up to 67 Hz for systems with up to 168 states.
  • The resulting reachable tubes are tighter than those from baseline methods, reducing conservativeness.
  • Formal guarantees on robust constraint satisfaction are preserved for both nonlinear analytic and neural network dynamics.
  • Solve times are reduced compared to previous approaches while maintaining real-time performance.

Where Pith is reading between the lines

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

  • The differentiability of the LEBs could allow end-to-end learning of dynamics models that are directly compatible with robust control.
  • Application to real-world systems like autonomous vehicles or robotics with learned NN dynamics might become practical due to the speed.
  • Extensions to other uncertainty sets beyond zonotopes could further improve tightness in specific applications.

Load-bearing premise

The NN verifier-generated affine relaxations combined with local Jacobian corrections produce LEBs that remain both sound and sufficiently tight for the downstream robust control problem to remain feasible and non-conservative.

What would settle it

Demonstration of a nonlinear or NN system where the computed LEBs do not overapproximate the actual linearization error, causing the closed-loop system to violate constraints despite the formal guarantees.

Figures

Figures reproduced from arXiv: 2607.01203 by Anutam Srinivasan, Glen Chou, Jeffrey Fang, Keyi Shen.

Figure 1
Figure 1. Figure 1: (a): Robust tubes from GPUSLS-LEO for a neural T pusher system and disturbed rollouts for a rotation and a push trajectory against random disturbances. (b): MPC rollout using our method on real dynamics, successfully moving the T to the goal. (c): Robust tubes of x and y position showing tight tubes and all simulated rollouts staying within the tubes. true nonlinear dynamics and their LTV approximation. Ex… view at source ↗
Figure 2
Figure 2. Figure 2: (a): LEBs on the satellite system. (b): LEBs on the quadrotor sys￾tem. In both systems, our method achieves the tightest over-approximation. TABLE I RUNTIME (IN MS) COMPARISON OF LINEARIZATION BOUND METHODS. System Random Global IH-classic IH-classic-CORA Path-based Satellite 0.894 0.143 0.218 11570 0.361 Quadrotor 1.278 0.156 0.453 227200 0.936 TABLE II LINEARIZATION ERROR UNDER NOMINAL OPTIMIZATION VIA P… view at source ↗
Figure 3
Figure 3. Figure 3: Satellite (7D). (a): Tube sizes for our method, GPUSLS-LEO, com￾pared to baselines and the variant without linearization error. Our method is the least conservative, while minimally increasing tube size (b): Tubes for the first four state dimensions; our tubes capture 100% of sampled rollouts. gradient descent (PGD) within a local neighborhood of radius ϵopt to reduce the error interval width. We compare P… view at source ↗
Figure 5
Figure 5. Figure 5: Planar quadrotor (6D). (a): System navigating through an obstacle field over a 20 m trajectory with a planning horizon length of 7500. (b): Tube widths along the horizon. Despite the long horizon, the tubes remain bounded due to the GPUSLS-LEO controller’s optimization of tube sizes. obstacle field over a horizon of 40 steps [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: Coupled Quadrotor. Per-iteration runtime evaluation across increasing numbers of coupled quadrotors. GPUSLS scales favorably with state dimension, growing approximately logarithmically with state and control space. Path-Based Hessians (PBH) achieves low runtimes at small dimensions but scales more poorly as the state dimension increases. dynamics, where the pusher successfully completes the task. We note t… view at source ↗
Figure 7
Figure 7. Figure 7: Neural T-pusher (5D). (a): Tube size comparison across ablations of our method, showing nonzero-centered tubes and linearization error gradients reduce tube widths. (b): Rollouts of our method; all trajectories remain in the computed robust tubes. (c): Rollouts of GPUSLS. Without formally considering linearization error, rollouts leave the tube. demonstrating the method’s scalability to state dimensional￾i… view at source ↗
read the original abstract

This paper studies real-time robust optimal control for uncertain nonlinear systems, where linear time-varying (LTV) approximations make planning tractable but require sound linearization error bounds (LEBs) to guarantee robust constraint satisfaction. We develop tight, differentiable, GPU-parallel LEBs for LTV approximations of nonlinear and neural network (NN) dynamics. For analytic dynamics, we introduce path-based Hessian bounds that are tighter than standard interval methods. For NN dynamics, we derive certified LEBs using NN verifier-generated affine relaxations and local Jacobian corrections. We adapt a GPU-parallel system-level synthesis LTV-based robust control solver to be compatible with these LEBs by extending it to handle right-invertible disturbance matrices and non-zero-centered disturbance sets for tight zonotopic uncertainty propagation. Our method, GPUSLS-LEO, enables online optimization of robust feedback policies that account for linearization error, producing tight, formally verified reachable tubes. On complex nonlinear and NN dynamics up to 168 state dimensions, our method can compute robust control policies on the GPU at rates up to 67 Hz, reducing solve times and conservativeness relative to baselines while preserving formal guarantees and real-time performance.

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

1 major / 2 minor

Summary. The paper claims to develop GPU-parallel, tight, differentiable linearization error bounds (LEBs) for LTV approximations of nonlinear and neural-network dynamics. For analytic dynamics it introduces path-based Hessian bounds; for NN dynamics it uses NN-verifier affine relaxations plus local Jacobian corrections. These LEBs are integrated into an adapted GPU-parallel system-level synthesis (SLS) robust controller that is extended to right-invertible disturbance matrices and non-zero-centered zonotopes. The resulting GPUSLS-LEO method is reported to compute robust feedback policies at up to 67 Hz on systems with up to 168 states while preserving formal guarantees and reducing conservativeness relative to baselines.

Significance. If the claimed soundness and tightness of the LEB constructions hold, the work would enable real-time, formally verified robust control for high-dimensional nonlinear and NN systems, a capability with clear relevance to robotics and autonomous systems. The explicit algorithmic constructions, GPU-parallel implementation, and solver extensions for zonotopic uncertainty are concrete strengths that could be adopted by the community.

major comments (1)
  1. [NN dynamics LEBs] Abstract and NN-dynamics LEB paragraph: the central claim that verifier-generated affine relaxations combined with local Jacobian corrections yield sound and sufficiently tight LEBs for downstream SLS propagation is load-bearing; the manuscript must supply the explicit error-bound derivation (including how the corrections interact with non-zero-centered zonotopes) so that soundness can be verified without gaps.
minor comments (2)
  1. [Abstract] The abstract states that the method reduces conservativeness relative to baselines; a dedicated table or figure with quantitative error-volume or feasibility comparisons (including interval-method baselines) should be referenced in the main text.
  2. [SLS solver adaptation] Notation for the adapted SLS solver (right-invertible disturbance matrices) should be introduced with a short equation or definition block before the algorithmic description to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for highlighting the importance of explicit soundness arguments for the NN-dynamics LEBs. We address the single major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [NN dynamics LEBs] Abstract and NN-dynamics LEB paragraph: the central claim that verifier-generated affine relaxations combined with local Jacobian corrections yield sound and sufficiently tight LEBs for downstream SLS propagation is load-bearing; the manuscript must supply the explicit error-bound derivation (including how the corrections interact with non-zero-centered zonotopes) so that soundness can be verified without gaps.

    Authors: We agree that an explicit, self-contained derivation is required for independent verification of soundness. In the revised manuscript we will expand the NN-dynamics LEB section (currently summarized in the abstract and the dedicated paragraph) to include the complete step-by-step derivation. This will (i) start from the verifier-generated affine relaxation of the NN, (ii) incorporate the local Jacobian correction term, (iii) propagate the resulting error set through the non-zero-centered zonotope representation, and (iv) show how the combined bound remains a valid over-approximation when fed into the extended SLS propagation. The added material will be placed immediately after the current high-level description so that readers can trace the soundness argument without external references. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained algorithmic construction

full rationale

The paper's central claims rest on explicit constructions (path-based Hessian bounds for analytic dynamics; NN verifier affine relaxations plus Jacobian corrections for NN cases) and extensions to an SLS solver for right-invertible disturbances and non-zero-centered zonotopes. These are presented as independent algorithmic steps benchmarked against external baselines, with no equations reducing performance metrics to quantities defined by fitted parameters from the same work, no load-bearing self-citation chains, and no renaming of known results as new derivations. The method preserves formal guarantees through sound LEB propagation without internal reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions from control theory and verification rather than new invented entities. No free parameters are explicitly fitted in the abstract description.

axioms (2)
  • domain assumption Dynamics admit bounded Hessians or Lipschitz Jacobians along trajectories (implicit in path-based Hessian bounds).
    Required for the analytic-dynamics LEB construction to be finite and computable.
  • domain assumption NN verifiers produce sound affine relaxations (standard in NN verification literature).
    Invoked for the NN-dynamics LEBs.

pith-pipeline@v0.9.1-grok · 5765 in / 1416 out tokens · 22213 ms · 2026-07-02T07:09:38.940613+00:00 · methodology

discussion (0)

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