P1 Models in Science and Engineering

Updated 19 November 2025

P1 Models are a diverse set of constructs defined across finite element analysis, radiative transport, statistical network theory, neural approximations, astrophysics, and algebraic geometry.
They utilize methodologies such as piecewise-linear discretizations, first spherical harmonic closures, log-linear reciprocal network formulations, and reinforcement learning for large language models.
Their practical implications include improved simulation stability, enhanced neural network approximation, refined statistical analysis of interactions, and optimized algorithms in both computational and experimental domains.

P1 Models encompass a wide range of technical constructs across mathematics, physical sciences, engineering, and computational disciplines. The term "P1" appears as a standard label for numerous model classes, including finite element spaces, statistical network models, physical defect types, special families of neural architectures, and more. The following entry surveys principal P1 models salient in contemporary research literature, drawing on the formal definitions, mathematical frameworks, algorithmic implications, and application domains established in leading arXiv contributions.

1. P1 in Finite Element and Numerical Analysis

"P1" most commonly denotes the space of globally continuous, piecewise-linear finite elements on simplicial meshes (affine-linear on each simplex). This class is fundamental for discretizing second-order PDEs, eigenvalue problems, and fluid mechanics:

Principal Eigenfunction Approximation: For second-order elliptic eigenproblems, the P1 finite element discretization, under suitable mesh conditions (simplicial, acute or Delaunay in an appropriate metric, interiorly connected), yields a stiffness matrix that is an irreducible M-matrix. This preserves positivity and simplicity of the discrete principal eigenvalue and strict sign of the eigenfunction (mimicking the continuous Perron–Frobenius property). The mesh requirements are sharp: acuteness in the D⁻¹-metric ensures all off-diagonal entries are nonpositive, guaranteeing M-matrix structure and thus spectral properties (Huang, 2013).
Stokes Equations with Slip Boundary Condition: The P1/P1 or P1b/P1 discretizations, combined with a penalty term to weakly impose slip boundary conditions, yield robust a-priori error estimates. Specifically, the energy-norm error for velocity and pressure satisfies O(h^{1/2} + ε^{1/2} + h/ε^{1/2}), where h is meshwidth and ε the penalty parameter. In 2D, reduced-order quadrature sharpens convergence to O(h + ε^{1/2} + h^{2/ε^{1/2})} (Kashiwabara et al., 2015).

2. P1 Approximation in Radiation Hydrodynamics

The P1-approximation in radiation hydrodynamics refers to the closure of the angular dependence in the radiative transport equation by the first spherical harmonic expansion (retaining only the isotropic and vector flux moments):

Navier–Stokes–Fourier–P1 Model: The compressible Navier–Stokes–Fourier set is coupled to the angularly truncated radiative transfer system. The unknowns are fluid variables and the first two radiation moments (I₀, I₁). The P1 system is written as:

$\epsilon \, \partial_t I_0 + \nabla \cdot I_1 = \theta^4 - I_0, \quad \epsilon\, \partial_t I_1 + \nabla I_0 = -I_1$

with coupling to the fluid via source and stress terms. In the non-relativistic limit (ε=1/c→0), the system reduces to an elliptic constraint for I₀, yielding a radiative diffusion correction in the energy balance, $-\operatorname{div}(\nabla I_0)$ (Jiang et al., 2015). In the low-Mach, non-equilibrium regime, the NSF-P1 model exhibits singular terms from radiation pressure, necessitating equivalent variables and delicate energy estimates. The parameter δ∈[0,2], scaling the scattering intensity, governs whether the limiting behavior is parabolic (diffusive), elliptic, or hyperbolic (wave-like) in the radiation (Li et al., 2022).

3. P1 Statistical Network Models

The directed random graph "p1 model" describes the statistical structure of dyadic interactions, with reciprocation, in social and network science:

Holland–Leinhardt p1 Model: Each dyad can assume four states: no edge, $i \rightarrow j$ , $j \rightarrow i$ , or reciprocal. The model is specified in log-linear form:

$\log p_{ij}(a,b) = \lambda_{ij} + a \alpha_i + b \beta_j + (ab) \rho_{ij}$

where $\alpha_i$ and $\beta_j$ encode node-level sender/receiver effects and $\rho_{ij}$ models reciprocation. The model is toric, with the algebraic structure captured by a multi-homogeneous toric ideal, and the existence of the MLE is characterized via the marginal polytope or its associated marginal cone (Rinaldo et al., 2010).

4. P1-KAN: Neural Networks via Kolmogorov–Arnold Representations

P1-KAN denotes a neural network architecture inspired by the Kolmogorov–Arnold superposition theorem. These networks utilize compositions of univariate, piecewise-linear (P1) splines in a layered structure:

Kolmogorov–Arnold Decomposition: Any continuous $f: [0,1]^n \to \mathbb{R}$ can be written as

$f(x) = \sum_{i=1}^{2n+1} \psi_i \left(\sum_{j=1}^n \Phi_{i,j}(x_j)\right)$

where $\Phi_{i,j}$ and $\psi_i$ are continuous univariate functions. P1-KAN generalizes this via L layers of width $n_\ell$ , utilizing learned piecewise-linear splines for each coordinate, with learnable knot placement via soft-sorting mechanisms. Notably, P1-KAN achieves superior approximation and numerical stability on irregular functions compared to MLPs and other KAN variants, especially at moderate spline resolutions ( $M \sim 20$ ) and depths ( $L=2,3$ ) (Warin, 2024).

5. P1 Models in Astrophysics and Defect Physics

In stellar population astrophysics and solid-state defect modeling, "P1" refers to populations or defect types:

Extended P1 in Globular Clusters: The photometrically defined "extended P1" stars in globular clusters such as NGC 2808 exhibit constant C, N, O, Na, Mg, Al abundances, inconsistent with a CNO-cycle nucleosynthetic origin. Spectroscopic studies constrain the origin to potential p–p chain nucleosynthesis, which would raise He without altering heavier elements, implying multiple enrichment channels within clusters (Cabrera-Ziri et al., 2019).
P1 Centers in Diamond: P1 centers are substitutional nitrogen defects with unpaired electrons. In type Ib diamond, clustered P1 centers with strong exchange couplings ( $J_{ij}\gtrsim 80$ MHz) exhibit broad, asymmetric EPR lineshapes and dominate DNP enhancements through truncated cross-effect pathways. Their spatial distribution and cluster fraction ( $f_c\gtrsim 23\%$ ) are key determinants of NV-center decoherence and DNP agent design (Bussandri et al., 2023).

6. P1 Variants in Mathematical and Algebraic Geometry

"P1-bundles" represent a geometric class of fiber bundles with projective line fibers, and the classification of such structures over projective manifolds of Picard number one yields a finite list of rational homogeneous spaces:

Double P1-bundle Theorem: For smooth projective $n$ -folds $X$ with rank-2 bundle $E$ such that $Z = \mathbb{P}_X(E)$ admits a second $\mathbb{P}^1$ fibration, only three cases exist (after normalization): flags of type $A_2$ ( $(\mathbb{P}^2,T_{\mathbb{P}^2})$ ), type $B_2$ ( $(\mathbb{P}^3,\mathcal{N})$ ), and $G_2$ ( $Q^5,\mathcal{C}$ ). These correspond to rational homogeneous manifolds $A_2$ , $B_2$ , $G_2$ ; the classification is established through numerical invariants (Chern classes, Fano indices), slope and discriminant computations, and bundle stability (Watanabe, 2012).

7. P1 Family of LLMs

P1 also designates a family of physics reasoning LLMs developed via reinforcement learning exclusively:

P1 LLMs for Physics Olympiad Reasoning: The P1 series comprises transformer-based models (30B and 235B parameter scales) that employ group-sequence policy optimization (GSPO), adaptive expansion, and pure-RL post-training on a curated Olympiad physics dataset (5,065 problems). The models demonstrate gold-medal performance on IPhO 2025 and competitive transfer to math and code tasks, further enhanced by agentic "PhysicsMinions" orchestration. Model ablations highlight the criticality of rule-based verifiers in stable RL (Chen et al., 17 Nov 2025).

This summary demonstrates the diversity and depth of "P1 models" in modern research, spanning from numerical analysis and PDE discretization, algebraic network statistics, and geometric classification, to neural approximation theorems, defect physics, stellar astrophysics, and AI research. Each instance requires precise mathematical definition and context, and implications for theory, computation, and experiment are highly domain-dependent.