P-NAVEM: Methods for Scientific, Robotic, & Naval Systems

• P-NAVEM is a multifaceted concept that incorporates neural network methods in scientific computing, energy-aware power management in autonomous robotics, and constrained 3D routing in naval engineering.
• The neural-based approach uses a tailored MLP with loss functions ensuring exact polynomial reproduction and C⁰ continuity, achieving optimal first-order convergence on complex polygonal meshes.
• Its robotics and naval implementations utilize branch-and-cut heuristics and adaptive DVFS techniques, balancing performance, energy efficiency, and compliance with constructibility and safety constraints.

P-NAVEM refers to several advanced methodologies spanning scientific computing, naval design, autonomous robotics, and power management, each underpinned by distinct algorithmic and modeling foundations. This entry systematically presents the leading interpretations and implementations of P-NAVEM documented in recent arXiv literature—specifically, as a neural-based extension of the Virtual Element Method for PDEs, as a power-management control system for autonomous robotics, and as a branch-and-cut-based multicommodity network flow engine for 3D pipeline routing in naval architecture.

1. Neural-Based P-NAVEM in Scientific Computing

P-NAVEM, "Polynomial Neural Approximated Virtual Element Method," designates a neural network-based variant of the Virtual Element Method (VEM), aimed at constructing globally continuous, low-order basis functions on arbitrary polygonal meshes without recourse to interior harmonicity constraints. In contrast to the standard NAVEM or B-NAVEM approaches, P-NAVEM enforces exact polynomial reproducibility and C⁰ continuity across elements via tailored loss functions and a boundary-lifting operator (Berrone et al., 14 Jan 2026).

Mathematical Formulation

For a given mesh element $E$ (convex or star-shaped polygon), with vertices $\{v_j\}_{j=1}^n$, P-NAVEM defines each local basis function via

$$\phi_{(j,E)}^p(z) = \mathcal{B}_{(j,E)}[\mathcal{N}^p](z) = \psi^0_E(z)\mathcal{N}^p(x_0) + \psi_{(j,E)}(z),$$

where $\psi^0_E$ is a bubble vanishing on $\partial E$, $\psi_{(j,E)}$ is a transfinite interpolant matching the VEM linear trace, and $\mathcal{N}^p$ is a fully connected network with inputs encoding point $z$, polygonal geometry, and basis index. The construction guarantees C⁰ conformality and—crucially—exact linear polynomial reproduction, enforced through an $L^2(E)$ moment-matching loss:

• Partition of unity: $\|\sum_j\phi_{(j,E)}^p - 1\|_{L^2(E)}$
• Linear reproduction: $\|\sum_j v_{j,i}\phi_{(j,E)}^p - x_i\|_{L^2(E)}$ for $i=1,2$

A gradient-matching penalty controls oscillatory derivatives.

Training and Deployment

The P-NAVEM MLP is trained offline per polygon class (up to seven sides), using a two-stage optimization (Adam, then L-BFGS) on dense quadrature points. Once trained, basis evaluations and gradients at quadrature points proceed via forward passes plus analytical differentiation, used for local matrix assembly in finite element PDE solvers.

Continuity and Computational Properties

Because the lifting operator aligns traces exactly with the VEM edge hats, P-NAVEM basis functions are globally C⁰ without extra degrees of freedom at element interfaces. Memory requirements remain moderate, and no duplication of edge data is needed.

Performance Characteristics

• Training time and memory scale as for the standard NAVEM approach; B-NAVEM is typically 8× slower due to Laplacian evaluation via autodiff.
• All methods reach optimal first-order convergence in $L^2$ and $H^1$ norms.
• On highly nonconvex or Voronoi-type meshes, P-NAVEM demonstrates 20–30% lower error constants relative to VEM and NAVEM, especially beneficial in PDEs with strong geometric heterogeneity.
• For nonlinear operators (e.g., variable-diffusion PDEs), P-NAVEM achieves speed-ups of 4–6× over classical VEM by dispensing with element projector/stabilization matrices, as these are subsumed into the trained network.

2. P-NAVEM in Cyber-Physical Power Management for AMRs

P-NAVEM also denotes a power-management framework for Autonomous Mobile Robots (AMRs), integrating cyber (compute, perception) and physical (motors, actuators) subsystems for real-time, energy-aware navigation planning and execution (Liu et al., 25 Nov 2025).

System Architecture

• Physical subsystem: Motors, actuators, batteries, 2D LiDAR, and RGB/Depth sensors.
• Cyber subsystem: CPU/GPU compute platform (NVIDIA Jetson), SLAM (e.g., AMCL), object detection (YOLOv3), planners (global and DWA local).
• Custom P-NAVEM modules:
  • Power Predictor: millisecond-scale inference of drivetrain and embedded system power from control and compute signals.
  • Locality Checker: dynamically assesses navigation locality via camera FOV overlap, SLAM confidence, and plan adherence.
  • Collision Predictor: computes Time-To-Collision (TTC) from LiDAR/velocity data.
  • Coordinator: adaptively adjusts DVFS settings, control periods, and perception/SLAM rates for energy optimization without loss of safety.

Power Modeling

Let $P_{AMR}(t) = P_M(t) + P_{ES}(t)$, where $P_M$ arises from motor currents (modeled by a trained MLP on $(v, \omega)$), and $P_{ES}$ is predicted from CPU/GPU frequency sets via regression fits on microbenchmarks for each rail.

Power costs are injected into the DWA planner’s objective to directly trade off motion energy with typical criteria (heading, distance, speed):

$$G(v, \omega) = \sigma(\alpha\,\text{heading} + \beta\,\text{dist} + \gamma\,\text{vel} + \theta\,p(v, \omega))$$

with $\theta$ governing energy-sensitivity.

Locality-Aware Adaptation

Three-tiered locality checks enable opportunistic reduction of perception/control frequency:

• Data level: If the camera FOV overlap $\theta_{FOV} > 0.8$, detection frame rate can be throttled.
• Position level: High SLAM particle confidence ($1/[\mathrm{Var}(x)+\mathrm{Var}(y)]$) permits reduced pose update rates.
• Path level: Stable global/local plan deltas allow reductions in computation.

All locality metrics above nominal thresholds, and safe time-to-collision ($s_t$) allow power-saving mode with minimized CPU/GPU settings; performance mode is triggered immediately upon threshold breach.

Experimental Outcomes

• End-to-end power prediction exceeds $96\%$ accuracy.
• Power consumption reductions: $38.1\%$ vs. default, $31\%$ over utilization-aware DVFS baselines.
• Slight increase in navigation latency ($\sim1.75\%$), improved localization accuracy (up to $38\%$ orientation, $18\%$ position vs. baseline).
• Minimum safe time-to-collision is never violated; energy-vs-latency/accuracy trade-offs remain acceptable for indoor platforms.

3. P-NAVEM for 3D Pipeline Routing in Naval Engineering

P-NAVEM in the context of naval design is a methodology for fully automated three-dimensional spatial pipeline routing, driven by a multicommodity network flow (MCNF) integer linear program combining physical, geometric, and constructibility constraints (Blanco et al., 2021).

Model Formulation

Let $G=(V,A)$ be a 3D grid graph, with "virtual nodes" to model elbows. For $K$ services, binary variables $x^k_{ij}$ indicate commodity $k$ using arc $(i,j)$. The objective is to minimize total per-arc cost (length, elbows, proximity, bonuses):

$$\min \sum_{k \in K}\sum_{(i,j)\in A} c^k_{ij}x^k_{ij}$$

Subject to:

• Node-disjointness: $\forall i\in V$, $\sum_k \sum_{j:(i,j)\in A} x^k_{ij} \leq 1$
• Flow conservation per $k$
• Single-unit flow boundary constraints

Constructibility Constraints

• Clearance (Dist): For any physical arc $a$, ensure no $k'\neq k$ uses arc $a'$ within threshold $R^{kk'}$:

$$\sum_{k'\neq k}\sum_{a': d_{a,a'} < R^{kk'}} x^{k'}_{a'} \leq M^k_a (1-x^k_a)$$

• Elbow-to-elbow (Elbow): Prohibit simultaneous use of virtual arcs within minimum separation $D^k$:

$$x^k_a + x^k_{a'} \leq 1 \quad \forall a,a' \in A^v, d_{a,a'} \leq D^k$$

Obstacle-avoidance is built-in by omission of edges intersecting obstacles.

Solution Engine

P-NAVEM solves the MCNF with a branch-and-cut framework—initially omitting Dist/Elbow constraints, then dynamically adding violated cuts (lazy constraints) on detected infeasibilities. As in most instances only local arc-pairs violate constraints, this approach remains computationally tractable even though the potential number of constraints is $\mathcal{O}(|A|^2 |K|)$.

Matheuristics

Two matheuristic algorithms are provided:

• H1 (Dimensionality-Reduction): Route each service independently, restrict feasible region to boxes around preliminary paths, incrementally expanding until feasible, then solve.
• H2 (Decomposition-Based): Iteratively allocate penalties to arcs with conflicts, solving via a hybrid of parallel/clustered/sequential shortest-path enumeration and elbow-repair. Commodity subproblems are solved using Dijkstra and elbow-violation feedback.

Computational Results

• On synthetic grids (up to $4.6$M nodes, $1.3$M arcs, $12$ services), branch-and-cut solves all low-density cases in less than 2 hours, with heuristic MIP gaps under $5.73\%$ (H1) and $1.59\%$ (H2).
• In a real ship-cabin scenario ($73,407$ nodes, $282,202$ arcs, $10$ services), exact branch-and-cut failed to return a solution in $12$ hours but H1 solved a pruned problem in $132$ seconds and H2 in $94$ seconds (objective within $3.4\%$).
• Matheuristically reduced problems yield an order-of-magnitude reduction in MILP instance size; decomposition-based heuristics offer nearly optimal solutions within minutes.

Scalability and Flexibility

• Branch-and-cut with lazy constructibility separation yields optimality guarantees on moderate problems—large-scale problems require heuristics.
• Dimensionality reduction and shortest-path heuristics exploit local geometry and inter-service conflict structure, providing practical tractability without quality loss.

4. Comparative Methodological Analysis

Interpretation	Domain	Core Technique
Neural-based P-NAVEM	Scientific computing	MLP for VEM basis, $L^2$ loss
Power management P-NAVEM	Autonomous robotics	Real-time power/DVFS control
Pipeline routing P-NAVEM	Naval engineering	MCNF, branch-and-cut, heuristics

The neural variant focuses on meshwide continuity and PDE fidelity; the robotics system on joint cyber-physical power minimization; and the routing system on constrained 3D network flow.

5. Implications and Significance

P-NAVEM in each interpretation responds to domain-specific requirements for scalability, physical constraint awareness, and algorithmic efficiency:

• In scientific computing, P-NAVEM leverages network-based function approximation to bypass complex stabilization and achieves favorable accuracy-complexity trade-offs in challenging mesh regimes (Berrone et al., 14 Jan 2026).
• In robotics, P-NAVEM enables real-time energy savings without compromising operational safety or state estimation, advancing the endurance and autonomy of battery-constrained AMRs (Liu et al., 25 Nov 2025).
• In naval engineering, P-NAVEM provides a unified, extensible engine for spatially constrained pipeline design, integrating high-dimensional search with constructibility logic (Blanco et al., 2021).

These advances collectively illustrate the growing integration of machine learning, optimization, and domain-specific modeling in modern computational engineering systems.