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arxiv: 2606.12735 · v1 · pith:XSZALDYJnew · submitted 2026-06-10 · 💻 cs.LG

Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources

Manuel Reyna , Alexandre Tartakovsky This is my paper

Pith reviewed 2026-06-27 09:44 UTC · model grok-4.3

classification 💻 cs.LG
keywords Physics-Informed Neural NetworksRadial Basis FunctionsDirac DeltaWeak FormResidual Least SquaresGroundwater TransportInverse ProblemsPartial Differential Equations
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The pith

Weak-form integration of Dirac deltas lets RBF-RLS solve PDEs accurately while PINN residuals fail to reach zero.

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

The paper establishes that Physics-Informed Neural Networks cannot drive residuals to zero on PDEs containing Dirac delta sources even after weak-form integration, whereas Radial Basis Function Residual Least Squares consistently yields accurate forward and inverse solutions. The difference is demonstrated on linear transport equations for groundwater flow and rivers using synthetic data, noisy data, and real measurements. Neural Tangent Kernel theory is invoked to account for the contrasting convergence behavior between the two Residual Least Squares formulations. A reader would care because singular sources appear in many physical models, and reliable machine-learning solvers for such cases would reduce the need for ad-hoc smoothing approximations.

Core claim

The paper claims that integrating the weak-form equation allows direct treatment of Dirac delta terms without smooth surrogates; under this treatment RBF-RLS produces good forward and inverse solutions to transport problems while PINNs do not make residuals converge to zero, with the distinction explained by Neural Tangent Kernel theory.

What carries the argument

Weak-form integration of the Dirac delta inside the Residual Least Squares loss, applied once to PINNs and once to an RBF expansion.

If this is right

  • RBF-RLS can be applied directly to inverse problems that fit real-world measurements without introducing modeling error from delta approximations.
  • The same weak-form treatment extends to other linear PDEs that represent flow and transport in porous media.
  • NTK analysis supplies a diagnostic tool for choosing among different Residual Least Squares architectures when singular sources are present.
  • Forward solutions remain stable under both synthetic and noisy data once the delta is integrated exactly.

Where Pith is reading between the lines

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

  • Similar weak-form handling might improve other neural or kernel methods on problems with point sources or discontinuities.
  • The observed gap suggests testing whether hybrid RBF-neural architectures inherit the convergence advantages of pure RBF-RLS.
  • The result raises the question of whether the same integration step improves performance on nonlinear transport equations.

Load-bearing premise

Neural Tangent Kernel theory correctly predicts why residuals converge under RBF-RLS but not under PINNs after the same weak-form integration of the delta.

What would settle it

Run both PINN and RBF-RLS on the same one-dimensional linear transport equation with a single Dirac delta source, integrate the weak form once, and check whether the PINN residual norm actually reaches machine zero or remains bounded away from zero.

Figures

Figures reproduced from arXiv: 2606.12735 by Alexandre Tartakovsky, Manuel Reyna.

Figure 1
Figure 1. Figure 1: RBF–RLS forward solution of the ADE with an instantaneous point release. (a) Solution [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of RBF–RLS and PINN forward and inverse solutions of the ADE with a step [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: RBF–RLS inverse solution of the ADE with first order exchange to an immobile zone fitted [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: PINN results for the DE with point source forcing. a) Evolution of the losses in PINN. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS formulations other than PINN, we focus on the Radial Basis Function (RBF) expansion (also known as a single-layer RBF Network). We show that while integrating out the Dirac delta in PINNs causes residuals to fail to converge to zero, RBF-RLS consistently provides good forward and inverse solutions to transport problems. We explain this finding using the Neural Tangent Kernel (NTK) theory. We test both approaches on linear PDEs that represent groundwater flow and transport in porous media and rivers. We solve inverse problems to fit synthetic data, noisy synthetic data, and real-world measurements.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims that standard PINNs, when Dirac delta sources are handled by integrating the weak-form residual, fail to drive the training loss (residual) to zero, whereas RBF expansions within the same RLS framework succeed at both forward and inverse solutions for linear transport PDEs; the distinction is attributed to properties of the Neural Tangent Kernel, with numerical demonstrations on groundwater flow and river transport models using synthetic, noisy, and real measurements.

Significance. If the empirical distinction and its NTK grounding hold after verification, the work would identify a concrete regime where deep PINN residuals stagnate on singular sources while shallow RBF-RLS remains effective, offering guidance for method selection in applications with point sources. The inclusion of inverse problems on real data strengthens applicability; explicit spectral analysis of the NTK (if added) would constitute a useful theoretical contribution.

major comments (2)
  1. [§4] §4 (NTK explanation): the assertion that NTK theory accounts for PINN residual non-convergence versus RBF-RLS success after weak-form delta integration is not supported by explicit computation of the relevant NTK eigenvalues or gradient-flow spectrum for the integrated loss operator; without this, the explanation cannot be distinguished from confounding factors such as network depth (deep vs. single-layer) or optimizer behavior.
  2. [Results] Results section (convergence plots and tables): the central claim that PINN residuals fail to reach zero while RBF-RLS succeeds requires demonstration that identical weak-form integration, quadrature, and collocation strategy are applied to both methods; any post-hoc differences in these implementation choices would undermine the attribution to the RLS formulation class.
minor comments (2)
  1. [§2] §2 (weak-form definition): clarify the precise quadrature rule used to integrate the Dirac delta term and confirm that the same rule is stated for both PINN and RBF-RLS.
  2. [Figures] Figure captions: add explicit labels for which curves correspond to PINN versus RBF-RLS and report the final residual norms numerically rather than only visually.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful comments, which help clarify the contributions and limitations of our work. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§4] §4 (NTK explanation): the assertion that NTK theory accounts for PINN residual non-convergence versus RBF-RLS success after weak-form delta integration is not supported by explicit computation of the relevant NTK eigenvalues or gradient-flow spectrum for the integrated loss operator; without this, the explanation cannot be distinguished from confounding factors such as network depth (deep vs. single-layer) or optimizer behavior.

    Authors: We acknowledge that our manuscript does not include explicit computation of NTK eigenvalues or the gradient-flow spectrum for the weak-form loss. The NTK explanation in §4 is based on established properties from the literature regarding the behavior of deep networks versus shallow linear models under least-squares losses. Specifically, deep PINNs can suffer from spectral bias and poor conditioning when the loss involves integrated singular terms, while the RBF expansion corresponds to a kernel method with more favorable properties for this setting. The comparison intentionally contrasts the standard deep PINN architecture with the single-layer RBF, as these represent typical choices in each RLS class. We will revise the text in §4 to emphasize that the NTK provides a qualitative rationale rather than a quantitative proof, and to explicitly discuss the role of network depth as part of the method distinction. No additional numerical NTK analysis will be added, as it would require substantial new theoretical work beyond the paper's scope. revision: partial

  2. Referee: [Results] Results section (convergence plots and tables): the central claim that PINN residuals fail to reach zero while RBF-RLS succeeds requires demonstration that identical weak-form integration, quadrature, and collocation strategy are applied to both methods; any post-hoc differences in these implementation choices would undermine the attribution to the RLS formulation class.

    Authors: The manuscript applies the same weak-form integration and RLS framework to both methods, as described in Sections 2 and 3. The quadrature and collocation points are chosen identically for the integrated residual in both cases, with the only difference being the parametrization of the solution (neural network weights versus RBF coefficients). To address this concern, we will add explicit statements and possibly a comparison table in the Results section confirming that the integration scheme, number of quadrature points, and collocation strategy are the same for PINN and RBF-RLS experiments. This will be supported by reference to the shared code repository if available. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation or claims

full rationale

The paper's core claim rests on an empirical observation that weak-form integration of Dirac deltas causes PINN residuals to fail to converge while RBF-RLS succeeds, with NTK theory invoked as an external explanatory framework. No equations, parameters, or results are shown to reduce by construction to fitted inputs, self-definitions, or self-citation chains; the distinction is presented as a tested difference rather than a tautological renaming or forced prediction. The derivation chain is self-contained against external benchmarks and does not rely on load-bearing self-references for its validity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that PINNs are equivalent to residual least-squares methods and that NTK theory applies to the observed residual behavior; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption PINNs can be interpreted as Residual Least Squares (RLS) methods
    Invoked to justify direct weak-form treatment of Dirac deltas without smoothing
  • domain assumption Neural Tangent Kernel (NTK) theory explains the difference in residual convergence between PINNs and RBF-RLS
    Used to account for why one method succeeds and the other fails after weak-form integration

pith-pipeline@v0.9.1-grok · 5730 in / 1411 out tokens · 23634 ms · 2026-06-27T09:44:36.910448+00:00 · methodology

discussion (0)

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Reference graph

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