arxiv: 2604.03371 · v1 · submitted 2026-04-03 · 💻 cs.RO

Recognition: no theorem link

Surrogate Model-Based Near-Optimal Gain Selection for Approach-Angle-Constrained Two-Phase Pure Proportional Navigation

Abhigyan Roy , Shreeya Padte , Abel Viji George , Vivek A , Satadal Ghosh

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:36 UTC · model grok-4.3

classification 💻 cs.RO

keywords two-phase pure proportional navigationoptimal gain selectionneural network surrogateapproach angle constraintguidance effortsurrogate modelnavigation gainsmissile guidance

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The pith

A neural network learns to predict near-optimal navigation gains for two-phase pure proportional navigation to achieve desired approach angles with minimal effort.

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

The paper shows how to select navigation gains in a two-phase version of pure proportional navigation so that a vehicle reaches a desired terminal approach angle while using the least total guidance effort. Because finding the best gains analytically is impossible for arbitrary starting positions and angles, the authors instead solve many numerical optimization problems to create training data. They then train a neural network to map any given initial and terminal geometry directly to the best gain pair. The network acts as a fast surrogate that avoids solving the optimization problem during flight.

Core claim

Because the optimal gains for the orientation and final phases of 2pPPN vary smoothly with the initial and desired terminal engagement geometries, a neural network regression model can be trained to predict those gains accurately. This model serves as a computationally efficient surrogate that generates near-optimal gain values on demand, allowing the two-phase guidance law to be realized with minimal total control effort for any specified approach angle within the feasible half-space.

What carries the argument

Neural network regression model trained to map engagement geometries to the optimal pair of navigation gains for the two phases of 2pPPN.

If this is right

The trained network enables real-time selection of near-optimal gains without solving optimization problems online.
Guidance systems can achieve the desired approach angle while minimizing the integrated guidance effort across both phases.
Multiple feasible trajectories in the orientation phase can be exploited to reduce overall control usage.
The approach generalizes to arbitrary initial and terminal conditions where analytical solutions do not exist.

Where Pith is reading between the lines

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

Similar surrogate models could be built for other multi-phase guidance laws where parameters vary smoothly.
In deployment, the network might allow faster response times for interceptors or autonomous vehicles.
Extending the training data to include noise or varying speeds could make the model more robust to real-world uncertainties.

Load-bearing premise

The relationship between engagement geometries and optimal gains is smooth enough that a neural network can interpolate and generalize accurately from a finite set of numerically optimized examples.

What would settle it

Collect new engagement geometries outside the training distribution, compute the true optimal gains by numerical optimization, and check whether the network's predictions match those optima within a small error tolerance.

Figures

Figures reproduced from arXiv: 2604.03371 by Abel Viji George, Abhigyan Roy, Satadal Ghosh, Shreeya Padte, Vivek A.

**Figure 2.** Figure 2: Angular half-space covered by 2pPPN D. Problem Statement Note from Section II-C that while the notion of extending a standard PPN guidance to a two-phase PPN guidance facilitated expanding the set of achievable approach angles to the entire angular half-space that depends on θ0 and αP0 , the choice of end configuration of Orientation phase trajectory, as presented in [14] and [15], in terms of αP and θ was… view at source ↗

**Figure 3.** Figure 3: In this angular plane, the slope of the line joining two (θ, αP) points represents the corresponding PPN navigation gain. Hence, reaching the desired terminal configuration P d f ≜ (α d Pf , αd Pf ) with bounded lateral acceleration requires approaching Pf along a line with a slope ≥ 2 (i.e., Nf ≥ 2). Thus, the Orientation phase drives the engagement configuration to this line on (θ, αP) plane. These cons… view at source ↗

**Figure 4.** Figure 4: Total cost variation v/s Nori for different αP0 , α d Pf , and Nf values. In addition to Nori and Nf bounds given in Section III-A, constraint on tf is also imposed to ensure practical mission feasibility. Without such a restriction, the optimization problem in (7) may allow trivial solutions in which the pursuer executes extremely slow orientation phase maneuvers with arbitrarily small lateral accelerati… view at source ↗

**Figure 5.** Figure 5: Optimal gain surfaces for different engagement ge [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 6.** Figure 6: directly predicts the optimal pair of navigation gains corresponding to the optimization problem (10b). Therefore, the neural network predicts the following relationship. FB : (αP0 , αd Pf ) → (N ∗ ori, N∗ f ) (15) This model has 2 inputs neurons corresponding to the engagement geometry, 3 hidden layers with 32, 64 and 16 neurons each, and 2 output neurons corresponding to (N∗ ori, N∗ f ). (N* f,N * ori) N… view at source ↗

**Figure 7.** Figure 7: Predicted vs. True Optimal Gains (Test Set) [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 8.** Figure 8: NN Predicted Optimal gain surfaces for different [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

read the original abstract

In guidance literature, Pure Proportional Navigation (PPN) guidance is widely used for aerodynamically driven vehicles. A two-phase extension of PPN (2pPPN), which uses different navigation gains for an orientation phase and a final phase, has been presented to achieve any desired approach angle within an angular half-space. Recent studies show that the orientation phase can be realized through multiple feasible trajectories, creating an opportunity to select navigation gains that minimize overall guidance effort. This paper addresses the problem of near-optimal gain selection for given initial and desired terminal engagement geometries. Two optimization problems are considered: i) determination of the optimal orientation-phase gain for a specified final-phase gain, and ii) simultaneously determining the optimal gain pair for both phases that minimizes the total guidance effort. Determining the optimal gains analytically for arbitrary engagement geometries is intractable. Numerical simulations further reveal that these optimal gains vary smoothly with respect to the engagement conditions. Exploiting this property, a neural network (NN)-based regression model is developed in this paper to learn the nonlinear mapping between optimal gains and initial and desired terminal engagement geometries. The trained NN serves as a computationally efficient surrogate for generating the optimal gains manifold, enabling near-optimal realization of 2pPPN guidance. Numerical simulation studies demonstrate that the developed NN-based architecture predicts optimal gains with high accuracy, achieving very high (close to 0.9) value of coefficient of determination.

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

NN surrogate for 2pPPN gains gives R² near 0.9 but stops short of showing the predicted gains actually deliver near-optimal total effort in closed-loop runs.

read the letter

The main takeaway is that the authors train a neural net to map engagement geometries to near-optimal gains for two-phase PPN under approach-angle constraints, and the net achieves decent accuracy on the gains themselves. They first observe that the optimal gains vary smoothly, generate labels via numerical optimization, then fit the regressor as a fast online replacement for that optimization. This is a practical move for anyone who wants real-time 2pPPN without solving the effort-minimization problem from scratch each time. The work is new in the narrow sense that it applies the surrogate idea specifically to the two-phase version with the angle constraint; the underlying PPN literature is cited and the smoothness observation is backed by their simulations. The R² close to 0.9 is a reasonable result for this kind of nonlinear mapping and shows the data-generation step was stable enough to learn from. The soft spot is exactly the one the stress-test flagged: accuracy on the gain values does not automatically translate to accuracy on integrated effort, because the mapping from gains to total effort is nonlinear and can be sensitive near the boundary of feasible angles. The paper reports no direct closed-loop comparison of total effort using NN gains versus the numerically optimal gains, so the “near-optimal” label rests on an assumption that has not been checked in the results shown. Details on how the training labels were produced, the optimizer used, and the train-validation split are also missing, which makes it harder to judge how robust the surrogate really is. This is the sort of paper that belongs in a guidance-and-control venue. Readers who work on missile or vehicle navigation laws would get value from the concrete implementation path it offers. It deserves a serious referee because the core engineering step is clear and reproducible in principle, even though the validation needs to be tightened on the actual performance metric. I would send it out with a request for those closed-loop effort numbers and a clearer data-generation section.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a neural network surrogate to select near-optimal navigation gains for two-phase pure proportional navigation (2pPPN) guidance. It formulates two optimization problems (fixed final-phase gain and joint gain-pair optimization) to minimize total guidance effort for specified initial and terminal geometries, generates training data via numerical simulations, and trains an NN regressor on the observed smooth mapping from engagement parameters to optimal gains, claiming R² values near 0.9 that enable real-time near-optimal 2pPPN.

Significance. If the surrogate truly delivers near-optimal closed-loop effort, the approach would allow computationally efficient gain selection for 2pPPN across varying initial conditions, which is valuable for real-time guidance of aerodynamically driven vehicles. The exploitation of smoothness in the optimal-gain manifold is a pragmatic strength, and the provision of machine-generated training data for regression is a positive methodological feature.

major comments (2)

[Abstract and Numerical simulation studies] Abstract and Numerical simulation studies section: The central claim of 'near-optimal realization of 2pPPN guidance' rests on R² ≈ 0.9 for gain prediction, yet no quantitative comparison of integrated guidance effort (or any other closed-loop performance metric) between NN-predicted gains and the numerically optimized reference gains is reported. Because the mapping from gains to total effort is nonlinear and potentially sensitive near approach-angle boundaries, an R² of 0.9 on the gains themselves does not establish that residual prediction errors preserve near-optimality of the effort.
[Optimization problems and data generation] Data-generation procedure (optimization problems section): No details are supplied on the numerical optimizer, convergence tolerances, number of samples, or train/validation/test split ratios used to label the optimal gains. Without these, the reliability of the reported R² and the generalization of the surrogate cannot be assessed.

minor comments (1)

[Problem formulation] Notation for the two optimization problems (i) and (ii) should be introduced with explicit symbols for the effort functional J and the gain variables before the NN architecture is described.

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 will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and Numerical simulation studies] Abstract and Numerical simulation studies section: The central claim of 'near-optimal realization of 2pPPN guidance' rests on R² ≈ 0.9 for gain prediction, yet no quantitative comparison of integrated guidance effort (or any other closed-loop performance metric) between NN-predicted gains and the numerically optimized reference gains is reported. Because the mapping from gains to total effort is nonlinear and potentially sensitive near approach-angle boundaries, an R² of 0.9 on the gains themselves does not establish that residual prediction errors preserve near-optimality of the effort.

Authors: We agree that the R² metric on gain prediction alone is insufficient to fully establish near-optimality of closed-loop guidance effort, given the nonlinear relationship between gains and total effort and possible sensitivity near boundaries. To address this limitation, we will add a quantitative comparison in the revised Numerical simulation studies section, reporting the integrated guidance effort (and relative error) achieved with NN-predicted gains versus the numerically optimized reference gains, including evaluation near approach-angle boundaries. revision: yes
Referee: [Optimization problems and data generation] Data-generation procedure (optimization problems section): No details are supplied on the numerical optimizer, convergence tolerances, number of samples, or train/validation/test split ratios used to label the optimal gains. Without these, the reliability of the reported R² and the generalization of the surrogate cannot be assessed.

Authors: We acknowledge that the absence of these implementation details limits assessment of reliability and generalization. In the revised manuscript, we will expand the Optimization problems and data generation section to specify the numerical optimizer (algorithm and implementation), convergence tolerances, total number of generated samples, and the train/validation/test split ratios used. revision: yes

Circularity Check

0 steps flagged

No significant circularity: standard surrogate modeling from independent optimizations

full rationale

The paper generates optimal gain values by solving two separate numerical optimization problems over families of engagement geometries. It then trains a neural network regression model on the resulting dataset to approximate the mapping from geometries to gains. The reported coefficient of determination (~0.9) quantifies the NN's predictive accuracy on held-out data relative to the pre-computed optima. This constitutes a conventional data-driven surrogate pipeline with no self-definitional loop, no renaming of a fitted quantity as an independent prediction, and no load-bearing self-citation that collapses the central claim. The derivation chain (optimize gains numerically → train NN → deploy NN for near-optimal gains) remains self-contained and externally falsifiable via the original optimization procedure.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that optimal gains vary smoothly enough for NN interpolation, plus the use of numerical optimization to generate training labels; no new physical entities are introduced.

free parameters (1)

Neural network architecture and hyperparameters
Number of layers, neurons, learning rate, and training epochs chosen to fit the mapping from engagement geometries to optimal gains.

axioms (1)

domain assumption Optimal gains vary smoothly with initial and terminal engagement geometries
Invoked to justify training a regression model that generalizes across arbitrary conditions.

pith-pipeline@v0.9.0 · 5576 in / 1264 out tokens · 53112 ms · 2026-05-13T18:36:58.315745+00:00 · methodology

discussion (0)

Reference graph

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