Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's

Avetik Arakelyan; Rafayel Barkhudaryan

REVIEW 4 major objections 4 minor 35 references

Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that PINN minimizers converge with probability one to the unique viscosity solution of fully nonlinear degenerate elliptic PDEs under a Hölder condition on the training samples.

desk verdict Theorem 4's leap from training-set convergence to a viscosity solution is not justified; the paper is a clean but flawed extension of [31]. read the letter →

arxiv 2501.04013 v1 pith:WRQSFSRL submitted 2024-12-19 math.NA cs.NAmath.AP

Avetik Arakelyan , Rafayel Barkhudaryan This is my paper · ORCID

classification math.NAcs.NAmath.AP MSC 65M1268T0741A4635J2535K20

keywords PhysicsInformedNeuralNetworksConvergenceViscositySolutionsFullynonlinearPDEsDegenerateellipticPINNHölderregularityDifferentialEquations

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 attempts to prove that Physics-Informed Neural Networks (PINNs) converge when applied to fully nonlinear degenerate elliptic PDEs. It claims that, as the number of training points grows, the sequence of network minimizers of the PINN loss converges uniformly, with probability one, to the unique viscosity solution of the PDE, provided the equation satisfies the comparison principle and the minimizers keep a uniform Hölder bound on the sampled points. If correct, this extends the existing convergence theory of PINNs beyond linear elliptic and parabolic equations to genuinely nonlinear equations such as Hamilton–Jacobi–Bellman equations.

What carries the argument

The main mechanism is the Hölder-regularized empirical PINN loss (16): the usual residual and boundary losses plus penalty terms $\lambda^R_{r,m}[F[h]]^2_{\alpha;U}$ and $\lambda^R_{b,m}[h]^2_{\alpha;\Gamma}$, with regularization coefficients that tend to zero as $m$ grows. The proof also relies on the probabilistic space-filling lemma (Lemma 1), which guarantees that random samples form an epsilon-net with high probability. Given the sample-bounded Hölder condition (19), the closest-point projection $v_{m_r}(x)=h_{m_r}(p_{m_r}(x))$ converts that sample bound into uniform equicontinuity on all of $U$, allowing Arzelà–Ascoli to extract a limit; degenerate ellipticity then passes the uniform residual convergence $F[h_{m_r}]$ to zero into the viscosity inequalities for the limit.

What would settle it

Check whether the largest value of $|h_{m_r}(x)-h_{m_r}(p_{m_r}(x))|$ over points $x$ in $U$ tends to zero along the proof's subsequence. The displayed argument gives uniform convergence of the projected functions $v_{m_r}=h_{m_r}(p_{m_r}(x))$ and of $h_{m_r}$ on the training set, while the theorem's conclusion is uniform convergence of $h_{m_r}$ on $\overline{U}$; bounding that difference is the step that would settle the proof, and an explicit example where the difference does not vanish would falsify the theorem as stated.

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Extended reading notes

Core claim

The central result is Theorem 4. Under the space-filling assumptions on the sampling distributions, with $m_r=O(m_b^{d/(d-1)})$, and under the condition $\sup_{m_r} [h_{m_r}]_{\alpha; U_{m_r}}<+\infty$, any subsequential limit of the minimizers $h_{m_r}$ of the Hölder-regularized PINN loss is a viscosity solution of $F[u]=0$ in $U$, $u=g$ on $\partial U$; uniqueness from the comparison principle then forces the whole sequence to converge to $u^*$ in $C^0(\overline{U})$ with probability one. The proof obtains compactness by projecting each minimizer onto the training sample set, uses the space-filling lemma to make the projected functions equicontinuous, and then passes the nonlinear operator through a viscosity-solution argument.

Load-bearing premise

The load-bearing premise is that the Hölder seminorm of each minimizer, evaluated only on the finite training sample, stays uniformly bounded as the sample grows; that sample-level bound is the only control the proof has over the minimizers between training points, and if it fails the compactness argument cannot start.

Editorial extensions

If this is right

The expected PINN loss at minimizers decays like $O(m_r^{-\alpha/d})$ once the sampling and Hölder parameters are fixed.
With probability one, the PDE residuals $F[h_{m_r}]$ converge uniformly to zero and the boundary values $h_{m_r}-g$ converge uniformly to zero.
The minimizers themselves converge uniformly on $\overline{U}$ to the unique viscosity solution, so the theorem supplies solution convergence and not merely loss convergence.
The result extends the earlier PINN convergence guarantee for linear equations to fully nonlinear degenerate elliptic equations that satisfy the comparison principle.

Reading between the lines

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

The proof as written establishes uniform convergence of the projected functions $v_{m_r}=h_{m_r}(p_{m_r}(x))$; uniform convergence of the minimizers themselves on all of $\overline{U}$ is the stated conclusion, but the displayed argument does not show it directly, so the theorem may be safest read as a statement about the projected minimizers together with convergence on the training sample set.
The sample-bounded Hölder condition (19) is a quantity one could try to enforce during training, for example by penalizing the Hölder or Lipschitz constant of the network on the training set, which would make the theorem's hypothesis testable in practice.
If condition (19) fails there is no compactness input for the viscosity argument, so the PINN loss could still go to zero while the networks oscillate between sample points; the condition is what separates loss convergence from solution convergence.
The same projection-and-compactness device could plausibly adapt to other operators with continuous residuals and a comparison principle, such as certain nonlocal or integro-differential equations, though the paper does not pursue that direction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the main theorem overreaches. The port of the Shin-Darbon-Karniadakis framework to fully nonlinear degenerate elliptic equations is clean, and Theorems 2 and 3 are solid as far as they go. But the step in Theorem 4 from convergence on the training set to u* being a viscosity solution does not work.

The problem is condition (19): it only bounds the Holder seminorm of h_mr on the sample set U_mr. The proof defines v_mr(x)=h_mr(p_mr(x)), which is piecewise constant, and shows v_mr converges uniformly. That tells you nothing about h_mr between samples. The assertion that following (22) there exists x_mr in U_mr where h_mr-phi attains a local maximum is unsupported; (22) only controls values at sample points. The stress-test example is right: take u*=-(x-1/2)^2 and h_mr=u*+(1/m)sin(2*pi*m*x). On the grid {k/m}, h_mr equals u*, so (22) and (19) hold, but h_mr-phi has no local maximum at any sample. So the viscosity-subsolution argument collapses, and the C^0 convergence claim is unproved.

To be fair, the paper does useful things. It correctly identifies that Schauder estimates don't transfer and sets up the comparison-principle machinery. The proofs of Theorems 2 and 3 are essentially [31] with F nonlinear, and they are written out cleanly. The notation is careful, and the limitation remarks are honest.

The fix is not cosmetic: you would need a global Holder (or other compactness) bound on the whole domain, not just on the samples. That would be a substantially stronger assumption.

Recommendation: this deserves a serious referee, because the question is important and the framework is plausible, but as written the central theorem is not proved. I'd send it to review with the expectation of major revision -- the referee should be asked to scrutinize the passage after equation (22). If the authors can supply a global compactness argument or a different route, the paper could become worth publishing; in its current form, no.

Referee Report

4 major / 4 minor

Summary. The paper studies convergence of Physics-Informed Neural Network (PINN) minimizers for second-order fully nonlinear degenerate elliptic PDEs. It adapts the probabilistic space-filling and Hölder-regularization framework of Shin, Darbon, and Karniadakis to the fully nonlinear setting. Theorem 3 asserts that minimizers of the regularized empirical loss have residuals F[h_mr] converging to 0 uniformly in C^0(U) and boundary values converging to g in C^0(Γ). Theorem 4 claims that, under an additional discrete Hölder bound (19), the minimizers converge uniformly on the closure of U to the unique viscosity solution u* of F[u]=0, u=g on Γ. The proof uses a step-function projection v_mr of h_mr onto the training set, uniform convergence of v_mr to u*, and a viscosity-stability argument.

Significance. If valid, Theorem 4 would be a valuable first general convergence result for PINNs in the fully nonlinear setting, and the paper is clearly organized, with appropriate use of viscosity solution theory and sample-filling estimates. The authors correctly identify that Schauder estimates cannot be used here. However, the central claim is not established: the proof of Theorem 3 contains a scaling inconsistency in the regularization rates, and the key step of Theorem 4—passing from convergence on the training set to a local maximum of h_mr−φ—is invalid under the stated assumptions. Moreover, Assumption 3 requires an exact zero-empirical-loss network in H_m for every m, which is stronger than universal approximation and is not justified. These are load-bearing issues, so the paper cannot be accepted in its present form.

major comments (4)

[Theorem 3 proof, line after Eq. (18)] The asserted rate $\hat\lambda^R_{r,m}=\hat\lambda^R_{b,m}=O(m_r^{-1/2-\alpha/d})$ is inconsistent with Eq. (13). Since $C_m=3\max\{\kappa_r\sqrt{d}^d m_r^{1/2},\kappa_b\sqrt{d}^{d-1}m_b^{1/2}\}$, the relation $m_r=O(m_b^{d/(d-1)})$ gives $C_m=\Omega(m_r^{1/2})$, so (13) yields $\hat\lambda^R_{r,m}=\Omega(m_r^{1/2-\alpha/d})$, which is not $o(1)$ for $d\ge2$ (it is $\Omega(1)$ when $d=2,\alpha=1$). Consequently the bound (18) cannot produce $Loss_m(h_m;\lambda,\lambda^R_m)=O(m_r^{-1/2-\alpha/d})$, and the claimed $Loss^{PINN}(h_m;\lambda)=O(m_r^{-\alpha/d})$ and the uniform decay $F[h_{m_r}]\to0$ are not established.
[Theorem 4 proof, paragraph after Eq. (22)] The claim that "Following (22) there exists a sequence of points x_mr in T... such that h_mr−φ attains its local maximum at x_mr" is unjustified. Equation (22) controls h_mr−u* only on the discrete set U_mr, and condition (19) bounds differences only for pairs in U_mr; neither controls h_mr between samples. For example, on U=(0,1), u*=-(x−1/2)^2, φ=0, and h_mr(x)=u*(x)+m_r^{-1}sin(2π m_r x), the training points k/m_r satisfy h_mr=u* on U_mr, so (22) and (19) hold, yet for large m_r the derivative of h_mr−φ at every sample point is −2(k/m_r−1/2)+2π, which is never zero; there is no local maximum at any sample. Thus the derivative conditions D(h_mr−φ)(x_mr)=0 and D^2(h_mr−φ)(x_mr)≤0 are not consequences of the stated assumptions, and the viscosity-subsolution step collapses.
[Theorem 4 proof, viscosity-subsolution paragraph] The proof only tests with functions φ satisfying ∇φ(x0)≠0. The viscosity subsolution property in Definition 2 must hold for every C^2 test function, including those with vanishing gradient. No argument is provided that the case ∇φ(x0)=0 follows by approximation, so the subsolution property is not fully proved even if the local-maximum step were valid.
[Assumption 3] The requirement that for each m the class H_m contain a network u*_m with Loss^{PINN}_m(u*_m;λ)=0 and with uniformly bounded Hölder seminorms of F[u*_m] is very strong and not justified. Universal approximation provides networks with small, not zero, empirical loss; the cited exact-representation results [10,11] cover special Hamilton-Jacobi equations, not the general fully nonlinear class considered here. Without an existence argument or an example of a family {H_m} satisfying Assumption 3, the theorem may be vacuous for the intended applications.

minor comments (4)

[Theorem 3 proof, displayed limit] The display '=λf ∫ ||G||^2...' should read λ_r∫_U ||G||^2 dμ_r(x_r), and the conclusion 'G=f' should be 'G=0', since f is not defined in that context.
[Theorem 2 proof] The proof invokes 'Lemma B.1 in [31]' without stating it; please include the lemma or its statement to make the paper self-contained.
[Theorem 4 proof] The variant of Arzelà-Ascoli applied to the discontinuous step functions v_mr is only cited to the appendix of [6]; it should be stated explicitly, since v_mr are not continuous.
[Throughout] There are several typographical errors: 'comparision' in Assumption 1, 'the Assumption 3' after Assumption 3, and the nonstandard apostrophe in 'PDE's' in the title and abstract.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorem is conditional on explicit assumptions and no step reduces the conclusion to its inputs.

full rationale

The derivation is self-contained in the sense required here: the convergence claims are not obtained by fitting a parameter and then predicting that same parameter, and no load-bearing premise is imported from the authors' own prior work. The paper's Assumptions 2 and 3 (sampling densities, an exact-zero-loss network in each class, uniform Holder bounds on the exact representatives) and condition (19) are stated as hypotheses, not as consequences of the theorem. The generalization argument of Theorem 2 is adopted from Shin, Darbon and Karniadakis [31] and applied to the fully nonlinear residual, and Theorem 4 then uses a separate compactness/projection argument. The known weakness in the proof—that (19) and (22) only control values on the training set, so the passage to a local maximum of h_mr - phi is unjustified—is a soundness gap in the mathematical argument, not a circular reduction of the conclusion to an input. No fitted quantity is renamed as a prediction and no self-citation carries the result.

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

The paper contributes no fitted parameters and introduces no new physical entities. Its central result rests on a chain of structural assumptions: degenerate ellipticity and comparison principle for the PDE, space-filling sampling, network classes containing exact zero-loss solutions, and a bounded Holder seminorm on the training set. The last assumption is the fragile one, because the proof inflates it into global uniform equicontinuity.

assumptions (5)

domain assumption Comparison principle for the fully nonlinear system (3)
Assumption 1; the entire result is conditional on uniqueness of viscosity solutions. This is a standard sufficient condition but restricts the PDE class.
domain assumption Degenerate ellipticity of F
Definition 1; used to compare F at the maximum points in the viscosity proof.
domain assumption Assumption 2 (sampling distributions fill the domain)
Borrowed from [31]; guarantees space-filling samples with positive probability.
domain assumption Assumption 3 (zero-loss network u*_m in H_m and uniform Holder bounds on F[u*_m] and u*_m at the boundary)
Requires the network class to contain an exact solution on every training set; often not satisfied by standard PINN architectures.
ad hoc to paper Condition (19): sup_mr [hmr]_{alpha;Umr} < infinity
The paper's extra compactness assumption. It is stated on the training samples only, but the proof uses it to obtain convergence on the whole domain, where it is not sufficient.

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Cite this review

Pith. "Pith review of Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's." pith.science (2026). https://pith.science/paper/WRQSFSRL

@misc{pith2026250104013,
  author       = {Pith},
  title        = {Pith review of: Convergence of Physics-Informed Neural Networks for Fully Nonlinear PDE's},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WRQSFSRL}},
  note         = {Machine review of arXiv:2501.04013}
}

read the original abstract

The present work is focused on exploring convergence of Physics-informed Neural Networks (PINNs) when applied to a specific class of second-order fully nonlinear Partial Differential Equations (PDEs). It is well-known that as the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We show that such sequence converges to a unique viscosity solution of a certain class of second-order fully nonlinear PDE's, provided the latter satisfies the comparison principle in the viscosity sense.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [31]

Communicat ions in Computational Physics 28(5), 2042–2074 (2020)

Shin, Y., Darbon, J., Karniadakis, G.E.: On the converge nce of physics informed neural networks for linear second-order elliptic and parabolic type pdes. Communicat ions in Computational Physics 28(5), 2042–2074 (2020)

work page 2020

[2] [1]

Baker, N., Alexander, F., Bremer, T., Hagberg, A., Kevrek idis, Y., Najm, H., Parashar, M., Patra, A., Sethian, J., Wild, S., AL., E.: Workshop report on basic rese arch needs for scientiﬁc machine learning: Core technologies for artiﬁcial intelligence. Tech. rep., USDO E Oﬃce of Science (SC), Washington, DC (United States) (2019)

work page 2019

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Neurocomputing 317, 28–41 (2018)

Berg, J., Nystr¨ om, K.: A uniﬁed deep artiﬁcial neural net work approach to partial diﬀerential equations in complex geometries. Neurocomputing 317, 28–41 (2018)

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Caﬀarelli, L.A., Cabr´ e, X.: Fully nonlinear elliptic eq uations, American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Pro vidence, RI (1995)

work page 1995

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Acta Mechanica Sinica 37(12), 1727–1738 (2021)

Cai, S., Mao, Z., Wang, Z., Yin, M., Karniadakis, G.E.: Phy sics-informed neural networks (pinns) for ﬂuid mechanics: A review. Acta Mechanica Sinica 37(12), 1727–1738 (2021)

work page 2021

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SIAM Journal on Mathematics of Data Science 1(4), 780–812 (2019)

Calder, J.: Consistency of lipschitz learning with inﬁni te unlabeled data and ﬁnite labeled data. SIAM Journal on Mathematics of Data Science 1(4), 780–812 (2019)

work page 2019

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SIAM Journal on Numerical Analysis 53(1), 82–104 (2015)

Calder, J., Esedoglu, S., Hero, A.O.: A pde-based approac h to nondominated sorting. SIAM Journal on Numerical Analysis 53(1), 82–104 (2015)

work page 2015

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Crandall, M.G., Evans, L.C., Lions, P.L.: Some propertie s of viscosity solutions of Hamilton- Jacobi equations. Trans. Amer. Math. Soc. 282(2), 487–502 (1984). https://doi.org/10.2307/1999247, http://dx.doi.org/10.2307/1999247

work page doi:10.2307/1999247 1984