Reconstructing Meson Parton Distributions from Mellin Moments with Artificial Neural Networks

Technical Summary: Reconstructing Meson Parton Distributions from Mellin Moments with Artificial Neural Networks

Problem Statement
The paper addresses the inverse problem of reconstructing a meson parton distribution function (PDF) or parton distribution amplitude (PDA), denoted as $q(x)$, from a finite set of Mellin moments $\{M_n\}_{n=0}^N$. The moments are defined as $M_n = \int_0^1 x^n q(x) dx$. This reconstruction is inherently ill-posed in the Hadamard sense: the mapping from a finite set of moments to the pointwise $x$-dependence is neither unique nor stable. Infinitely many functions can share the same first $N$ moments, and small perturbations in the input data can induce large, unphysical oscillations or spurious multiple maxima in the reconstructed profile. Traditional approaches often rely on rigid parametric ansätze (e.g., Beta-like forms) which introduce significant functional bias, or entropy-based methods that impose additional selection criteria.

Methodology
The authors propose a flexible, non-parametric reconstruction strategy using Artificial Neural Networks (ANNs) constrained by first-principles physics.

1. Physical Constraints (Hard and Soft):
   Instead of assuming a specific functional form, the method enforces known physical properties of meson distributions:

   • Hard Constraints (Architectural): Implemented via "wrappers" that guarantee exact satisfaction of:
     • Support: $x \in [0, 1]$.
     • Normalization: $\int_0^1 q(x) dx = 1$.
     • Endpoint Behavior: $q(0) = q(1) = 0$, achieved by multiplying the network output by a prefactor (e.g., $x(1-x)$) or symmetrizing the output.
     • Symmetry: For pions (and quarkonia), exact $x \leftrightarrow 1-x$ symmetry is enforced.
   • Soft Constraints (Loss Penalties): To suppress artifacts typical of ill-posed inversions (spurious bumps and curvature reversals), differentiable penalties are added to the loss function:
     • Unimodality ($L_{uni}$): Penalizes slopes that violate a single-peak structure.
     • Concavity ($L_{conc}$): Penalizes positive curvature (inflection points) near the peak.

2. Network Architectures:
   Two interchangeable neural cores are compared to approximate the underlying function $f(x)$ before the physical wrappers are applied:

   • Multilayer Perceptron (MLP): A standard feedforward network with fixed nonlinearities on nodes and trainable linear weights on edges.
   • Kolmogorov–Arnold Network (KAN): A network where trainable nonlinearities (splines) reside on the edges, and nodes perform simple summation. The KAN utilizes B-splines with compact support.

3. Training Procedure:
   The network is not trained on pointwise data (which is unavailable) but on the moments. The loss function minimizes the weighted least-squares difference between the target moments and the moments predicted by the network. Crucially, the predicted moments are computed via numerical integration (quadrature) of the network's pointwise output on a fixed mesh. This integration step is differentiable, allowing gradients to flow back to the network parameters.

Key Results
The method was benchmarked against synthetic target distributions (both symmetric "pion-like" and asymmetric "kaon-like") where the exact function and moments were known.

• KAN vs. MLP Performance: The KAN consistently outperformed the MLP in reconstructing the pointwise $x$-profile, particularly for asymmetric distributions (kaons) and when the number of input moments was small ($N=5$ vs. $N=10$).
  • Both networks reproduced the input moments with comparable accuracy, demonstrating the ill-posed nature where different profiles can fit the same moments. In some asymmetric cases (e.g., Table 2), the MLP was even slightly more precise for specific moment orders $n$.
  • However, the KAN showed systematically lower relative errors in the lowest orders ($n=1, 2, 3$) across all three benchmarks (symmetric and asymmetric), which determine the global location, width, and skewness of the distribution.
  • The KAN produced profiles that were visually and statistically closer to the true target function, despite the comparable moment reproduction.
• Role of Locality: The authors attribute the KAN's superiority to the locality of its B-spline basis. In a KAN, adjusting a spline coefficient affects the function only within a small window of $x$. In contrast, MLP weights have global support; a gradient step to correct the fit in one region can induce spurious global oscillations elsewhere. This locality allows the KAN to "sculpt" the asymmetric peak without destabilizing the rest of the domain.
• Data Scarcity: When trained on only 5 moments, the KAN maintained high fidelity in the low-order moments and the central region, whereas the MLP reconstructions deviated more significantly from the true profile.

Significance and Claims
The paper claims to demonstrate a method for reconstructing meson distributions with minimal functional bias. By replacing rigid parametric ansätze with flexible neural cores constrained only by rigorous physical principles (support, normalization, symmetry) and statistical expectations (unimodality), the approach avoids the "hard bias" of assuming a specific shape family.

The authors argue that the KAN architecture is particularly well-suited for this class of inverse problems due to its structural locality, which naturally dampens the global oscillations that plague standard MLPs in ill-posed moment inversions. The work serves as a feasibility study, establishing that neural networks can accurately recover the shape of distributions from limited moment data. The authors note that while the current "fit-per-dataset" approach is computationally intensive, future work will explore amortized inference (training a single model on a library of synthetic moments) to enable rapid inference, though this would trade the explicit hard-constraint enforcement for implicit learning from the training data distribution.

The paper concludes that while the KAN's advantage is likely due to the specific inductive bias provided by the spline-based, edge-localized architecture rather than raw capacity, the KAN used in the study possessed substantially more parameters than the MLP. The authors explicitly state that capacity alone could in principle account for part of the advantage, though they do not believe this is the primary explanation, and they call for a controlled comparison at matched parameter counts to rigorously isolate the effect of the spline basis.