arxiv: 2605.04768 · v1 · submitted 2026-05-06 · 📡 eess.SY · cs.SY

Recognition: unknown

From open-loop representations to closed-loop feedback implementations in differential games: A numerical case study

Philipp Braun , Timothy L. Molloy , Gal Barkai , Iman Shames

Authors on Pith no claims yet

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

classification 📡 eess.SY cs.SY

keywords differential gamesfeedback strategiesneural networkssurveillance-evasionopen-loop solutionssample-and-holdpursuit-evasion

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The pith

Neural networks trained on open-loop solutions can approximate discontinuous feedback strategies for surveillance-evasion games.

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

The paper develops a numerical method to convert open-loop representations of solutions to the prying-pedestrian surveillance-evasion game into closed-loop feedback strategies. Open-loop trajectories supply training data for neural networks that learn to map current states directly to player controls. This matters because open-loop solutions are easier to compute for many pursuit-evasion and surveillance games, yet real applications require feedback that responds to new information or disturbances. Simulations demonstrate that networks trained with a loss function tied to the game's payoff objectives produce effective strategies. The work also measures the performance gap that arises when these strategies are implemented via sample-and-hold rather than continuous updates, which stems from jumps in the optimal feedback.

Core claim

Feedback strategies for the prying-pedestrian surveillance-evasion differential game can be realized as neural-network input-output maps trained on state-control pairs drawn from open-loop solution data. These maps reproduce the desired closed-loop behavior in simulation when the training loss is chosen to respect the players' objectives. Because optimal feedback is discontinuous, the paper further shows how to quantify the payoff loss each player experiences when replacing continuous-time feedback with its sample-and-hold counterpart.

What carries the argument

A neural network that approximates the feedback strategy as a state-to-control mapping, trained by supervised learning on open-loop trajectory data.

If this is right

Feedback strategies become available for games where only open-loop solutions have been computed.
The same training procedure applies to other pursuit-evasion and surveillance-evasion problems.
Sample-and-hold implementations of the learned strategies incur a measurable performance loss that can be bounded.
Neural networks can represent the mapping even when the underlying feedback contains discontinuities.

Where Pith is reading between the lines

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

The method could let robotic systems or autonomous vehicles adopt game-theoretic controllers without repeatedly solving partial differential equations online.
Adding robustness terms to the training loss might produce strategies that remain effective under sensor noise or model mismatch.
The approach might extend to multiplayer games with more than two participants once open-loop data can be generated at scale.

Load-bearing premise

Open-loop solution data is rich enough to let the neural network learn the true feedback map, including across surfaces where the strategy jumps.

What would settle it

Compare the payoff achieved when both players use the trained network feedback against the payoff of the known open-loop optimum, for many random initial states; a large systematic gap would falsify the claim that the approximation recovers the closed-loop solution.

Figures

Figures reproduced from arXiv: 2605.04768 by Gal Barkai, Iman Shames, Philipp Braun, Timothy L. Molloy.

**Figure 1.** Figure 1: Coordinate system attached to the evader. view at source ↗

**Figure 2.** Figure 2: Neural network used to obtain approximations of view at source ↗

**Figure 3.** Figure 3: Approximation of the optimal value function view at source ↗

**Figure 4.** Figure 4: Approximation of the gradient of V in terms of ψdV,1(ξ) on the left and ψdV,2(ξ) on the right view at source ↗

**Figure 5.** Figure 5: Feedback law ψu,1(ξ) (left) and ψu,2(ξ) (right). where κ10(x) = 10|x|1 + |x| 2 2 denotes a linear combination of the 1-norm and the 2-norm. The data for the training is generated by simulating openloop representations of solutions obtained through the method described in Braun et al. (2025) in backwards time and for parameters selected as ρ = 1, ve = 1.5, vp = 1 and we = 1. The method returns data of the … view at source ↗

**Figure 7.** Figure 7: Solutions starting at ξ0 = [0, 1]⊤ using the sampleand-hold feedback (21)-(22) for different sampling periods. From left to right, the sampling periods (23) are used. The games end after time units in (24). and the pursuer can only update their input at discrete time steps. We define the set-valued map representing the time to end the game with the sample-and-hold dynamics (21)-(22) as V δe δp : S ⇒ R≥0. … view at source ↗

**Figure 8.** Figure 8: Visualization of V δ min(·) (left) and V δ max(·) (right) for different parameter selections δ ∈ {0.05, 0.1, 0.2} (from top to bottom). a piecewise constant input signal for t ∈ [0, δ]. Thus, the optimal value functions V δ min(ξ0) and V δ max(ξ0) may differ from V (ξ0) even in the case where the optimal feedback laws are single valued. While this paper only investigates the impact of sample-and-hold contr… view at source ↗

read the original abstract

Solutions to pursuit-evasion and surveillance-evasion differential games are typically computed and expressed using open-loop representations, with the synthesis of feedback strategies significantly less common. We propose a numerical scheme for obtaining feedback strategies for the recently introduced prying-pedestrian surveillance-evasion differential game. The scheme involves computing feedback strategies as input-output maps approximated via neural networks trained using data obtained from open-loop representations of solutions. Simulations show the effectiveness of neural networks trained with an appropriate learning-loss function. Since optimal feedback strategies are discontinuous, as a second contribution, the potential loss/gain of individual players is subsequently studied for players using sample-and-hold feedback compared to continuous-time feedback.

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

NNs trained on open-loop data give usable feedback approximations for this one game in simulation, but the evidence stays thin on metrics and discontinuity coverage.

read the letter

This paper takes open-loop trajectories for the prying-pedestrian game and trains a neural network to map states to controls, then checks how sample-and-hold versions of those controls perform against ideal continuous feedback. The approach is straightforward and directly tackles the gap between theory and implementable strategies that the field often leaves open. Generating training data from solved open-loop instances and picking a loss that respects the game structure is a practical move, and the sample-and-hold comparison adds a concrete check on real-world discretization effects. That part is useful because discontinuities in optimal feedback are common in these games and cannot be ignored in code. The simulations are presented as evidence that the method works, which is fair for a case study. The soft spot is the lack of reported numbers. No trajectory errors, cost gaps, or success rates across repeated trials appear in the description, so it is hard to tell how close the closed-loop behavior stays to the true optimum. The concern about open-loop paths undersampling the switching surfaces is reasonable: those trajectories follow smooth segments and cross discontinuities at isolated points, so the data may leave thin coverage exactly where the feedback jumps. A continuous network could then interpolate poorly there and produce larger deviations than the abstract suggests. This is a targeted numerical study for people already working on surveillance-evasion or pursuit-evasion games who need a template for turning open-loop solvers into deployable feedback. It is concrete enough to deserve peer review, though the referees will likely ask for quantitative error tables and more detail on training-set density near the discontinuities.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a numerical scheme to synthesize feedback strategies for the prying-pedestrian surveillance-evasion differential game by training neural networks on state-control pairs obtained from open-loop solution representations. Simulations are presented to illustrate the effectiveness of this approach when an appropriate loss function is used, and a secondary analysis compares the performance loss/gain when players implement sample-and-hold feedback versus ideal continuous-time feedback in the presence of discontinuities.

Significance. If the central claim holds, the work supplies a practical, data-driven route from tractable open-loop computations to implementable closed-loop feedback in differential games, where direct feedback synthesis is often intractable. The explicit treatment of discontinuities via sample-and-hold comparisons is a constructive contribution. The absence of quantitative error metrics and coverage analysis in the reported simulations, however, leaves the effectiveness claim only partially substantiated.

major comments (2)

[Abstract] Abstract and simulation results: effectiveness is asserted via simulations, yet no quantitative metrics (e.g., trajectory deviation norms, value-function errors, or success rates across initial conditions), error bounds, or explicit description of how discontinuities are handled during training or closed-loop rollout are supplied. This directly weakens support for the claim that the trained networks produce feedback strategies whose closed-loop behavior matches the true optimal feedback.
[Section describing data generation and NN training] Data-generation procedure (open-loop trajectories used for training): trajectories are generated by solving the game from selected initial conditions and therefore consist of smooth arcs punctuated by isolated switches. No analysis, sampling-density plots, or verification is given that these trajectories densely populate neighborhoods of the discontinuity surfaces (switching manifolds) that separate regions of qualitatively different optimal controls. Because the NN is a continuous approximator, gaps in this coverage would produce large interpolation errors precisely where the true feedback jumps, undermining generalization to closed-loop trajectories.

minor comments (2)

[Figures] Figure captions and axis labels in the simulation plots could more explicitly indicate which curves correspond to open-loop reference, NN-closed-loop, and sample-and-hold implementations.
[Training subsection] A brief statement of the precise loss function employed (beyond 'appropriate learning-loss function') would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the revisions planned for the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and simulation results: effectiveness is asserted via simulations, yet no quantitative metrics (e.g., trajectory deviation norms, value-function errors, or success rates across initial conditions), error bounds, or explicit description of how discontinuities are handled during training or closed-loop rollout are supplied. This directly weakens support for the claim that the trained networks produce feedback strategies whose closed-loop behavior matches the true optimal feedback.

Authors: We agree that the current presentation relies primarily on visual trajectory comparisons to illustrate effectiveness and lacks quantitative metrics. In the revised manuscript we will add quantitative measures, including average trajectory deviation norms and success rates over a grid of initial conditions, together with a clearer description of how discontinuities are approximated during training and how sample-and-hold is realized in closed-loop rollouts. These additions will provide stronger support for the claim that the learned feedback reproduces the optimal closed-loop behavior. revision: yes
Referee: [Section describing data generation and NN training] Data-generation procedure (open-loop trajectories used for training): trajectories are generated by solving the game from selected initial conditions and therefore consist of smooth arcs punctuated by isolated switches. No analysis, sampling-density plots, or verification is given that these trajectories densely populate neighborhoods of the discontinuity surfaces (switching manifolds) that separate regions of qualitatively different optimal controls. Because the NN is a continuous approximator, gaps in this coverage would produce large interpolation errors precisely where the true feedback jumps, undermining generalization to closed-loop trajectories.

Authors: The referee correctly notes the risk of insufficient coverage near switching manifolds. Although the selected initial conditions were chosen so that many trajectories cross the discontinuity surfaces, we did not supply density plots or explicit verification of neighborhood coverage. We will add a sampling-density analysis and visualization in the data-generation section of the revision to confirm that the neighborhoods of the switching manifolds are adequately populated, thereby supporting the generalization properties of the continuous neural-network approximator. revision: yes

Circularity Check

0 steps flagged

No significant circularity: open-loop data trains NN approximator without reducing claims to fitted inputs by construction

full rationale

The paper's central contribution is a numerical scheme that first solves the differential game in open-loop form for selected initial conditions, then uses the resulting state-control pairs as supervised training data for a neural network that approximates the feedback map. This is standard data-driven approximation; the open-loop trajectories are generated independently via optimal control solvers and are not defined in terms of the NN outputs. The subsequent closed-loop simulations and sample-and-hold comparisons are empirical evaluations performed after training, not tautological predictions. Any self-citations pertain to the prior definition of the prying-pedestrian game and are not invoked to justify uniqueness or to force the NN result. No equation or derivation step equates the reported effectiveness to a parameter fitted from the same data used to claim that effectiveness. The method therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The approach assumes open-loop solutions exist and can be sampled densely enough to train generalizable feedback maps; no explicit free parameters, axioms, or invented entities are stated in the abstract.

pith-pipeline@v0.9.0 · 5415 in / 981 out tokens · 17785 ms · 2026-05-08T17:05:11.205658+00:00 · methodology

discussion (0)

Reference graph

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