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REVIEW 5 major objections 5 minor 56 references

PINN-DG: Residual neural network methods trained with Finite Elements

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A discontinuous-Galerkin finite element loss for PINNs is proven to converge to the exact solution of Poisson's equation as the mesh and network refine in step.

desk verdict Genuinely useful FE-interpolated PINN training with DG stabilization, but the convergence proof has a repairable gap in Lemma 3.3 and sign errors that need fixing. read the letter →

arxiv 2507.03521 v2 pith:WIW2LS2U submitted 2025-07-04 math.NA cs.NAphysics.comp-ph

Georgios Grekas , Charalambos G. Makridakis , Tristan Pryer This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 65M1565M12
keywords physics-informedneuralnetworksdiscontinuousGalerkinfiniteelementtrainingresidualGamma-convergencePoissonequationdeepRitzliftingoperator
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a way to train physics-informed neural networks in which the loss is evaluated not by collocation at random points but by finite element interpolation and integration, in a discontinuous Galerkin formulation. The authors try to establish that this reformulation is not merely a computational shortcut: minimizers of the DG-interpolated residual energy converge to the true solution of Poisson's equation as both the mesh and the network approximation capacity improve in step. If true, this matters because the FE-based loss avoids costly pointwise evaluations of second derivatives of the network, inherits stability from the finite element framework, and comes with a rigorous convergence proof built from variational arguments. Numerical experiments support the claim by showing competitive accuracy, 3–11× speedups over collocation, large memory savings, and robustness on an L-shaped domain with a corner singularity.

What carries the argument

The load-bearing object is the DG-interpolated residual energy (3.1): $$E_\ell(u_\ell) = \sum_{K\in T_h} \int_K |\$\Delta$ I_S u_\ell + I_S f|^2\,dx - 2\sum_{e\in E^i_h} \int_e \{\!\{\$\Delta$ I_S u_\ell + I_S f\}\!\} [\![\nabla I_S u_\ell \cdot n]!]\,ds + \$\alpha$\,\text{pen}(u_\ell),$$ with $I_S$ the $C^0$ Lagrangian interpolant into piecewise polynomials of degree $q\ge2$, $\{\!\{\cdot\}\!\}$ the average, $[\![\cdot]!]$ the jump across interior edges, and $\text{pen}(u_\ell)$ the inverse-mesh-weighted jumps of gradients plus a boundary penalty realizing Dirichlet conditions in the spirit of Nitsche's method. Rewriting via the lifting operator $R_h$, the energy is $\int_\Omega |L_h(u_{h,\ell})+f_h|^2 - |R_h(\nabla u_{h,\ell})|^2 + \alpha\,\text{pen}(u_\ell)$, where $L_h = \Delta_h - R_h(\nabla u_h)$ is a discrete Laplacian. This object simultaneously removes pointwise network second derivatives — the loss is assembled from finite element integrals — and supplies the coercivity and compactness needed for the $\Gamma$-convergence (lim inf–lim sup) proof: the lifting is $L^2$-bounded by the penalty, the discrete Laplacian passes weak limits to the true Laplacian (Lemma 3.1), and the consistency and jump terms vanish on recovery sequences.

What would settle it

Run the unit-square experiment of Section 4.1 with the four-block residual network, fix a mesh schedule satisfying $h(\ell)=c\beta_\ell$, and record the $L^2$ error of the interpolated minimizers as the loss decreases; if the errors fail to approach zero while the discrete losses do, Theorem 3.1's convergence conclusion is not realized for that architecture, isolating either the approximation-rate premise or the scaling condition as the failing ingredient.

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Extended reading notes

Core claim

The central claim is that replacing the pointwise residual loss of a PINN with a discontinuous-Galerkin finite element loss does not sacrifice consistency: minimizers $u_\ell$ of the discrete energy $E_\ell$ converge, after $C^0$ finite element interpolation, weakly in $H^1$ and strongly in $L^2$ to the unique minimizer $u$ of the continuous residual energy $\int_\Omega |\Delta u + f|^2\,dx$, provided $h(\ell) = c\beta_\ell$ for a constant $c>0$ and the boundary-mesh condition $h_{E,\ell}^{-1/2}(\tilde\beta_\ell^{[2,0,\infty]})^{1-2\epsilon} \le C$ holds, where $\beta_\ell$ is the best network approximation rate from (2.18). Since the unique minimizer of $E$ is the weak solution of Poisson's equation, the theorem states that the DG-trained networks approximate the PDE solution in the limit.

Load-bearing premise

The load-bearing premise is that the neural network spaces actually contain functions that approximate the exact solution to the steadily improving degrees described in assumption (2.18), and that the mesh is refined in step with those rates; if the optimizer does not find such approximants, the $\Gamma$-convergence proof does not apply.

Editorial extensions

If this is right

  • For elliptic problems like Poisson's equation, training a PINN can be performed entirely through finite element assembly and quadrature, eliminating automatic differentiation of the network's second derivative in the loss.
  • The gradient-jump consistency terms carry the stability: without them, deeper residual networks show $L^2$ errors one to two orders of magnitude larger even as the reported loss decreases.
  • Weak imposition of Dirichlet data through the boundary penalty means the network architecture does not need to encode boundary conditions.
  • Any accumulation point of the interpolated discrete minimizers is the exact solution of the Poisson problem, so the convergence conclusion is independent of optimizer-specific behaviour once the approximation-rate hypothesis holds.
  • On nonconvex domains with corner singularities the FE-trained method still converges under refinement, with $L^2$ error decreasing from about $6\cdot10^{-2}$ to $9\cdot10^{-3}$ in the reported L-shaped test.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same DG-interpolation template should transfer to other second-order elliptic operators and to nonlinear least-squares residual energies whenever a coercive DG discretisation with a bounded lifting exists, since the proof mechanism is not specific to the Poisson operator.
  • The scaling condition $h(\ell)=c\beta_\ell$ could be turned into an adaptive algorithm: estimate the network's approximation rate during training and refine the mesh accordingly, and the theorem predicts a convergent schedule.
  • The jump-free failure mode suggests that pure residual PINN losses can be fooled by nearly discontinuous functions that quadrature misses, so adding a DG-style jump penalty to collocation-based PINNs may improve robustness, although the paper does not test this.
  • Extending to parabolic or hyperbolic problems would require a time-discrete version of the discrete Laplacian; the variational template indicates a route, but new coercivity and compactness estimates for the evolution operator would be needed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper introduces PINN-DG, a physics-informed neural network method for elliptic boundary value problems in which the network is interpolated into a C0 finite element space and a discontinuous-Galerkin-type discrete energy, including gradient jump consistency terms and a Nitsche-style penalty, is minimized. The central theoretical claim is a Gamma-convergence result: under an abstract neural-network approximation property and the scaling h(ell)=c beta_ell, minimizers of the discrete energy, after finite element interpolation, converge weakly in H1 and strongly in L2 to the unique minimizer of the continuous residual energy, which solves Poisson's equation. The paper also reports numerical experiments on square and L-shaped domains showing accuracy, robustness for deeper networks, and computational and memory savings relative to collocation-based PINNs. The main proof is built on an equicoercivity proposition and a lim inf / lim sup Gamma-convergence argument.

Significance. If completed, the paper gives a useful convergence framework for a practical hybrid method: replacing pointwise network derivatives with finite element interpolation and DG jump terms is a genuinely attractive way to reduce cost and memory in residual-based PINN training. The conditional structure of the theorem, based on abstract network approximation rates, is clearly stated and is compatible with standard neural-network approximation results. The numerical experiments support the practical value of the method, particularly the stabilizing role of the jump terms for deeper networks. However, the theoretical proof as printed has a load-bearing gap in the lim sup lemma, as well as several sign and regularity inconsistencies, so the central convergence claim is not fully established in the current version.

major comments (5)
  1. [§3.4, Lemma 3.3, Eq. (3.27)] The proof of the lim sup inequality is incomplete. The lemma's goal (3.27) contains two limits, but the estimates (3.28)-(3.35) address only the first: they bound the difference between ||Δw_{ell,delta}+f_ell||^2_{L2} and E_ell(w_{ell,delta}) and show that the consistency and penalty terms vanish. There is no argument establishing the second limit, ||Δw_{ell,delta}+f_ell||^2_{L2} -> E(w), along the diagonal delta=h(ell)^{1/4}, h(ell)=c beta_ell. A triangle-inequality estimate such as | ||Δw_{ell,delta}+f_ell||^2_{L2} - E(w) | <= C( beta_ell^{[3,2,2]} delta^{-2} + ||f_ell-f||_{L2} + delta ) would supply the missing step. Without it the recovery sequence is not shown to satisfy the lim sup inequality, and Theorem 3.1 is not fully established.
  2. [§3.4, Theorem 3.1 and Lemma 3.3, Eqs. (3.30)-(3.32)] The rates used in Lemma 3.3 are not controlled by the rates assumed in Theorem 3.1. The proof of (3.30)-(3.32) requires the approximation rate beta_ell^{[4,3,2]} appearing in (3.26) with s=4, i.e. ||w_{ell,delta}-w_delta||_{H^3} <= beta_ell^{[4,3,2]} |w_delta|_{H^4}. The definition of beta_ell in Theorem 3.1 is the maximum of beta_ell^{[1,0,2]}, beta_ell^{[2,1,2]}, beta_ell^{[3,2,2]}, and beta_ell^{[4,2,2]} only. Therefore the proof uses a rate that is not an assumption of the theorem; either beta_ell must be enlarged to include beta_ell^{[4,3,2]}, or the estimates in Lemma 3.3 must be reworked so that only the listed rates are used.
  3. [§3.4, Lemma 3.2, Eq. (3.24)] Equation (3.24) rewrites E_ell(v_ell) as ||L_{h(ell)}(bv_ell)+f_ell||^2_{L2} + ||R_h(∇bv_ell)||^2_{L2} + alpha pen(v_ell), but the definition in (3.6) contains -||R_h(∇bv_ell)||^2_{L2}. The subsequent inequality in (3.24) effectively uses the correct negative sign together with the lifting bound (3.8), but the displayed identity is wrong and must be corrected. The same issue occurs in the integration-by-parts identity (3.19), where the volume term sum_K ∫_K bv_ell Δbv_ell should carry a minus sign; the proof is insensitive to this because it estimates absolute values, but the identity as printed is incorrect.
  4. [§3.4, Theorem 3.1, boundary-mesh condition] The assumption h_{E,ell}^{-1/2} (beta_ell^{[2,0,∞]})^{1-2ε} <= C stated in Theorem 3.1 is never used in the proof. The boundary penalty estimate (3.33)-(3.35) is performed entirely with h(ell) and the rates beta_ell^{[1,0,2]} and beta_ell^{[2,1,2]}; no lower bound on boundary edge lengths or the parameter ε appears. Either the proof should show where this condition is needed, or the condition should be removed, since an unused hypothesis makes the convergence statement unnecessarily restrictive and unverifiable in the experiments.
  5. [§2.1, Proposition 2.1, Eq. (2.10)] The proof of estimate (2.9) contains an unjustified step. In (2.10), the bound (∑_{e∈E^i_h} h_e^{-1} ||f_h||^2_{L^2(K_e)})^{1/2} ≲ ||f_h||_{L^2(Ω)} is asserted, but for a generic piecewise polynomial f_h, already for f constant on a uniform mesh, the left-hand side is of order h^{-1/2}||f_h||, not bounded by a constant times ||f_h||. The statement of Proposition 2.1 also does not state the regularity of f, while the proof uses f ∈ C^0(Ω) and then invokes ||f||_{H^1(Ω)}. Since Lemma 3.3 uses (2.9) to show the consistency term vanishes, this estimate, or its hypotheses, must be repaired.
minor comments (5)
  1. [Proof of Theorem 3.1] The bound ||ũ - bu_ell||_{L^2(∂Ω)} ≲ ||ũ - bu_ell||_{L^2(Ω)} ||ũ - bu_ell||_{H^1(Ω)} is not the standard multiplicative trace inequality, which has square roots on the two norms; the conclusion is unaffected because both factors tend to zero, but the displayed estimate should be corrected.
  2. [§3.2, Lemma 3.1, Eq. (3.13)] In the chain of estimates (3.13), the constant factors from the uniform bound (3.10) are omitted, and the second term should carry the factor (∑_e h_e^{-1} ||J∇u_h K||^2)^{1/2}; as printed, an independent factor h |φ|_{H^1} appears without explanation.
  3. [§3.4, Lemma 3.3, Eq. (3.32)] The simplification h(ell)^2 δ^{-4} ( (beta_ell^{[4,3,2]})^2 δ^{-2} + 1 ) = h(ell) + h(ell)^{3/4} is not consistent with the substitution h(ell) = beta_ell and δ = h(ell)^{1/4}; the correct leading terms are of orders h^{5/2} and h. Since both tend to zero, this is cosmetic but should be fixed.
  4. [Proof of Theorem 3.1, Eq. (3.36)] The notation j_ell(ũ_ell) + b_ell(ũ_ell) in (3.36) is undefined; it should refer to the internal-edge and boundary parts of pen(u_ell), respectively.
  5. [§4, Computational Experiments] The numerical experiments report speedups and memory comparisons but do not state the number of independent runs or any error bars; adding this information would strengthen the robustness claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorem is a conditional Gamma-convergence argument with explicit approximation assumptions; the authors' self-citations are contextual and not load-bearing.

full rationale

The paper's central derivation is self-contained in the sense required by the circularity test: no quantity is defined in terms of the result being proved, no parameter is fitted to data and then renamed as a prediction, and the convergence theorem rests on explicitly stated assumptions rather than on a self-citation chain. The discrete loss (3.7) is constructed from finite element interpolation, DG consistency terms, and penalty terms; Theorem 3.1 then proves convergence of its minimizers under the abstract neural-network approximation assumption (2.18) and the scaling h(ell) = c beta_ell together with the stated boundary-mesh condition. These hypotheses are declared as assumptions in the text ('we now describe the approximation properties of neural network spaces. For each ell in N... there exists w_ell in V_ell satisfying (2.18)'), and they are not derived from the theorem's conclusion. Lemma 3.2 and Lemma 3.3 use standard Gamma-convergence estimates; technical support is drawn from external references such as Di Pietro-Ern, Buffa-Ortner, and Brenner-Scott, alongside the authors' prior work [24], [27], [29]. The self-citations situate the method within the authors' earlier frameworks, but the proof of Theorem 3.1 is re-derived rather than imported wholesale: Lemma 3.1 is proved in the text, Proposition 3.1 is proved in the text, and Lemma 3.3 supplies its own estimates. The only notable manuscript-level issue is a completeness gap, not circularity: in Lemma 3.3 the second limit in (3.27), namely |E(w) - ||Delta w_{ell,delta} + f_ell||^2_{L2}| -> 0, is asserted but not explicitly estimated; this is a missing proof step that would need to be supplied for the lim sup inequality to be fully rigorous as written. That is a correctness risk, not a reduction of the result to its inputs. Accordingly, no circular step is exhibited and the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The method does not fit constants to data, so the circularity burden is low. The proof rests on five explicit assumptions: C0 interpolation and trace estimates, DG lifting boundedness, abstract NN approximation rates, a grid-network coupling condition, and smooth approximation of V functions. The main concern is that the central theorem is conditional on rates that experiments do not verify.

free parameters (2)
  • Penalty parameter alpha = 60 in experiments; alpha > C_R in theory
    Chosen by hand. It balances the residual term against the gradient-jump and boundary penalty terms. The theory only requires alpha sufficiently large, with no optimal value derived.
  • Scaling constant c in h(ell) = c beta_ell = unspecified
    The convergence theorem requires the mesh size to be exactly proportional to the neural network approximation rate. The constant c is free and affects the convergence rate but is not fitted or estimated in the paper.
assumptions (5)
  • domain assumption Abstract neural network approximation property (2.18): for each ell there exists w_ell in V_ell approximating w in W^{s,p} with rate beta_ell tending to zero.
    The convergence proof relies on these rates for s up to 4. The paper cites approximation literature, but does not verify the rates for the networks and training used in the experiments.
  • standard math C0 Lagrange interpolation estimates (2.6) and trace estimates (2.7) hold on shape-regular triangulations.
    These are standard finite element interpolation results used throughout the proofs of Proposition 2.1, Lemma 3.1, and Lemma 3.3.
  • standard math The DG lifting operator R_h satisfies the L2 bound (3.8) from [13, 19].
    This bound is used to control the consistency term, prove compactness of the discrete Laplacian, and choose alpha > C_R in Lemma 3.2.
  • ad hoc to paper Scaling condition h(ell) = c beta_ell and boundary-mesh condition h^{-1/2}_{E,ell} (beta_tilde[2,0,infty])^{1-2epsilon} <= C in Theorem 3.1.
    This coupling between mesh size and neural network approximation rate is introduced specifically to make the Gamma-convergence proof close. It is stated as an assumption, not derived or tested.
  • domain assumption Elliptic regularity and smooth approximation of V functions with zero boundary data, as described in Remark 3.1.
    The recovery sequence in Lemma 3.3 uses smooth functions w_delta vanishing on the boundary, constructed by odd extension. This requires regularity and density results imported from Evans [22].

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Cite this review

Pith. "Pith review of PINN-DG: Residual neural network methods trained with Finite Elements." pith.science (2026). https://pith.science/paper/WIW2LS2U

@misc{pith2026250703521,
  author       = {Pith},
  title        = {Pith review of: PINN-DG: Residual neural network methods trained with Finite Elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WIW2LS2U}},
  note         = {Machine review of arXiv:2507.03521}
}
read the original abstract

Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical techniques such as finite elements and finite differences. In this paper, we introduce a new class of Physics-Informed Neural Networks (PINNs) trained using discontinuous Galerkin finite element methods. Unlike standard collocation-based PINNs that rely on pointwise gradient evaluations and Monte Carlo quadrature, our approach computes the loss functional using finite element interpolation and integration. This avoids costly pointwise derivative computations, particularly advantageous for elliptic PDEs requiring second-order derivatives, and inherits key stability and accuracy benefits from the finite element framework. We present a convergence analysis based on variational arguments and support our theoretical findings with numerical experiments that demonstrate improved efficiency and robustness.

Figures

Figures reproduced from arXiv: 2507.03521 by the authors.

Figure 1
Figure 1. Throughout this section, we use continuous piecewise polynomial finite element spaces of degree 2. The penalty parameter in the discrete energy functional (3.1) is fixed at α = 60. Recalling the residual neural network (ResNet) architecture defined in Equations (2.11), (2.13), and (2.14), we restrict our experiments to two-dimensional domains Ω ⊂ R 2 . The width of each hidden layer is fixed at N = 64, and we compar… view at source ↗
Figure 1
Figure 1. Uniform mesh: The unit square is divided into 3 × 3 = 9 squares, each further subdivided into 2 triangles. All neural network components are implemented in the PyTorch framework [45], with parameters stored in float32 precision. All computations were performed on an RTX A4500 GPU. 4.1. Approximating Solutions in Convex Domains. We begin by considering ap￾proximations on convex domains. Specifically, we choose the ri… view at source ↗
Figure 2
Figure 2. Comparison of L 2 -errors for solutions uθ obtained using a residual neural network with two blocks. We minimise the full energy functional (3.1) (solid lines) and a reduced variant without gradient jump terms (dashed lines). The error ||uθ − u||L2(Ω) is plotted against the number of mesh cells, where u is the exact solution (4.1). Green, orange, and yellow curves correspond to quadrature rules of precision 1, 2, an… view at source ↗
Figures from the paper (7 more)
Figure 3
Figure 3. Figure 3: As in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png]
Figure 4
Figure 4. Figure 4: Execution time ratio over 1,000 training iterations comparing the proposed finite element method against quadrature-based collocation, using a residual neural network with 2 blocks. As the number of mesh cells and integration precision increase, the finite element meth…
Figure 5
Figure 5. Figure 5: As in [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Memory usage during training with numerical integration of precision 1, comparing the proposed method using finite element inter￾polation (interpolant) against standard automatic differentiation (auto￾grad). Results are shown for residual neural networks with 2 and 4 b…
Figure 7
Figure 7. Figure 7: Memory usage during training with numerical integration of precision 2, comparing the proposed method using finite element inter￾polation (interpolant) against standard automatic differentiation (auto￾grad). Results are shown for residual neural networks with 2 and 4 b…
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Pointwise absolute error |uθ−u| for varying numbers of trian￾gles. Top row: 384 triangles (h = 1/4), 1536 triangles (h = 1/8); bottom row: 6144 triangles (h = 1/16), 24576 triangles (h = 1/32). A residual network with two blocks and numerical integration of first-order…

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