Physics-informed neural networks viewpoint for solving the Dyson-Schwinger equations of quantum electrodynamics

Comment: Matching state-of-the-art numerics

Reviewer:

“The PINN results do not match state-of-the-art numerical solutions, making the implementation seem meaningless unless it at least matches them.”

Reply: The author acknowledges that quantitative agreement with traditional numerical algorithms remains important, particularly since those methods often define the standard for validating PINN-based results. In the revised section IV, subsection A, the following improvements were introduced to address both viewpoints:

1. Reparametrization: The neural network now learns log B instead of B directly, which improves performance for small mass values by compressing the dynamic range to [log(10^-12), log(1)].
2. Boundary penalties: Loss terms have been added to guarantee B(1) = 0 and B(10^-12) ≈ 10^-3, securing the ends of the domain.
3. Differentiable integration: All NumPy-based code has been replaced with native JAX routines, ensuring backpropagation is preserved over the full computational graph.

As a result, the PINN now reproduces the benchmark solution for B. The improved performance is visualized in the updated Figure 1.

Comment: Use of NumPy vs. differentiable tools

Reviewer:

“The NumPy reliance is irrelevant; use TF or JAX integrators (e.g. MC or tensorized trapezoid).” Reply:

All calls to NumPy within training loops have been removed. The entire PINN training pipeline is now implemented using JAX with differentiable operations throughout. This change enables full gradient flow through the integral operators and lines up the author´s approach with best practices in differentiable programming.

Comment: What is the fundamental advantage of using a PINN?

Reviewer:

“Given all the approximations needed even for traditional solvers, what unique advantage does PINN offer? Can one drop any of those approximations entirely?”

Reply: In section IV, subsection B, the author emphasizes the potential strengths and limitations of the PINN framework:

• Continuity: The PINN produces a smooth, continuous approximation of B(p^2), rather than relying on a discrete grid, i.e., while traditional solvers in general produce solutions evaluated on a fixed momentum grid, the PINN learns a continuous function B, enabling smooth evaluation at arbitrary points without the need for interpolation.
• Architecture extensibility: The same network design may be extended to more complex systems, e.g., coupled photon-fermion DSEs, although these directions are still under investigation by the author for further works.
• Exact residual minimization: Automatic differentiation enables minimization of the residual integral equation at each training point, offering a flexible computational scheme.

However, the author is aware that, since validation against traditional numerical algorithms is typically expected by the established DSEs community, the absence of reliable benchmark solutions from the standard methods in certain regimes or extended set of equations may limit the ability to confirm the correctness of the PINN’s predictions.

Minor Points:

1. Definition of mass and coupling:

Reply: m and alpha are now defined immediately after Equations (3) and (4) in section II.

1. Notation in section 3.4:

Reply: To prevent confusion, the author defined the PINN output as B̃(x), reserving B(x) for the reference solution. The author also clarifies that the training is not supervised in the classical sense. As stated in section III, subsection C, the PINN learns by minimizing the residual between its own prediction and the integral operator applied to it, consistent with the standard PINN methodology.

1. Line styles in Figure 1 (previous version):

Reply: The updated figure 2 (the figure 1 of the previous version) all curves for different values of alpha appear superposed as a single line because, under the rainbow truncation in Landau gauge, the function A =1 is an exact solution of the DSE for the fermion propagator. This gauge/truncation choice removes the contribution of the vector part of the fermion self-energy, and thus A does not acquire any nontrivial momentum dependence. The identical curves reflect this known analytic behavior. We have clarified this point in the caption of figure 2 and throughout the text of section IV, subsection A. 4. Fixes to figure 4:

Reply: Figure 1 (figure 4 in the previous version) has been regenerated with the same axes, limits, and aspect ratio as figure 2 (previous version), showing excellent agreement and were included side-by-side.

1. Rewrite of pages 12–13:

Reply: These pages have been reviewed to detail the changes made to the model and loss, report new error metrics, and remove any language suggesting the PINN was ineffective. Finally, the author is grateful for the reviewer’s thoughtful comments, which have significantly improved the clarity, technical correctness, and relevance of the manuscript.

Comment 1: Rainbow truncation and generality

Reviewer:

“The numerics relies on the ‘rainbow approximation’ for treating the coupled integral equations. Does this restrict the generality of the PINN method?”

Reply: Indeed, the author adopts the rainbow truncation as a well-known first step that captures fundamental non-perturbative features, e.g., dynamical mass generation. However, the proposed PINN framework is expected to not be restricted to this approximation, but of course the author agrees that it is good research practice to proceed cautiously and investigate calmly this topic. As mentioned in sections III and V, in future works more sophisticated truncation schemes such as including vertex corrections or the coupled photon–fermion system will be considered and are expected to be incorporated directly into the loss function without modifying the principles of the neural network approach implemented in this work.

Comment 2: Gauge independence

Reviewer: “The choice of Landau gauge simplifies the equations, but the method should work independent of gauge—please comment.”

Reply: Landau gauge was selected to refine the presentation and simplify the leading-order behavior of A. Nonetheless, the method is in principle desired to be gauge independent. One can readily modify the loss function to include the appropriate integral expressions for A and B in other gauges. The same PINN architecture and training scheme apply. As mentioned in section V, the author is currently investigating for further work the possibility to study the unfamous gauge (in)dependence issue explicitly. However, it is worth noting that traditional numerical algorithms are generally gauge dependent when truncations are employed, unless specific care is taken to preserve gauge invariance, for instance, using Ward–Takahashi identities. Although the author is confident that a gauge-invariant PINN formulation could be developed under suitable conditions, the current expectation within the Dyson–Schwinger equations community has been that this novel method should first reproduce established numerical results, which are often regarded as “real” benchmarks. From this viewpoint, PINNs are expected to reflect the same gauge dependence as standard discretization-based approaches, without introducing additional artifacts beyond those arising from the chosen truncation.

Comment 3: Loss function and trivial solutions

Reviewer: “Please explain the choice of loss in Eq.~(22) and whether the PINN ever collapsed to trivial solutions (e.g. B = 0).” Reply:

The baseline loss in Equation (22) consists of a standard mean-squared error (MSE) between the predicted and target values of B. However, in very early experiments the network often collapsed to the trivial solution B ≈ 0, which is mathematically consistent but physically irrelevant. To mitigate this, the following measures were taken:

• A softplus activation was used to ensure positivity of B.
• The bias of the output layer was initialized to match B ~ 10⁻³ in the infrared.
• Additional loss components were introduced, targeting mid, high, and UV momentum intervals with log-scale penalties.

These refinements are detailed in Section III, subsections A and C, and were essential to recovering physically meaningful solutions that match traditional numerical results, especially the last one (which is the novelty of the present version of the work).

Comment 4: Figures and comparison with the traditional solver

Reviewer:

“Please merge Figs.~1 & 3 and Figs.~2 & 4 into side-by-side panels, improve the A(p²) presentation, and include a direct comparison to the standard numerical method.”

Reply: The figures have been updated accordingly. The author now presents side-by-side plots comparing the traditional numerical approach and the PINN results for both A and B (see Figures 1 and 2 in the revised manuscript). Axis labels and font sizes have been standardized for readability. Since A = 1 identically in Landau gauge, this fact is now emphasized in both the figure caption and the discussion in Section IV, subsection A. In the updated figure 2 (which replaces the figure 1 of the previous version) all curves corresponding to different values of alpha appear superposed as a single line. This is because, under the rainbow truncation in the above-mentioned gauge, the function A =1 is an exact solution of the DSE for the fermion propagator. This gauge/truncation choice removes the contribution of the vector part of the fermion self-energy, and thus A does not acquire any nontrivial momentum dependence. The identical curves reflect this known analytic behavior.

Finally, the side-by-side comparison clearly shows that the updated PINN framework, with improved loss components and initialization strategies, achieves excellent agreement with the benchmark traditional method over all momentum scales.