{"id":"33f12fad-1f3e-4cdf-92e4-fbfe3919808a","arxiv_id":"2501.04013","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Minimizers of the PINN loss are claimed to converge uniformly to the unique viscosity solution of degenerate elliptic fully nonlinear PDEs, under zero-loss capacity and sample-Hölder assumptions.","lead":"This paper claims that as the number of sample points grows, the neural networks minimizing a Physics-Informed Neural Network loss converge to the unique viscosity solution of a fully nonlinear PDE, provided the PDE satisfies the comparison principle. For a generalist, it is a theoretical convergence guarantee for a popular scientific machine learning method, but the assumptions include an exact zero-loss network and a Hölder condition on the samples.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4 never establishes uniform convergence of h_mr on U; the projected functions v_mr are step-like, and the local-maximum step only uses sample values, so the viscosity argument collapses.","rationale":"The reader's rejection is justified. Theorem 4 is the central claim, and its proof has a load-bearing gap: it establishes uniform convergence of the projected step functions v_mr, not of the neural-network minimizers h_mr. The later viscosity argument needs h_mr to have local maxima near x0, but this is only checked on the training set. Condition (19) bounds the Holder quotient of h_mr only on samples, so h_mr can oscillate between samples; the explicit one-dimensional example shows that even with (19) and (22) satisfied, hmr-phi need not have any local maximum at a sample point. Thus the proof of the subsolution property collapses. I am not disputing the plausibility of the high-level template or the contribution of adapting [31] to fully nonlinear equations; however, the stated theorem is not proved. Because my concern is the same gap identified by the reader, I do not adjust the verdict: the paper should not be accepted as written. A stronger global Holder condition on h_mr (or a different argument passing from v_mr to h_mr) would be needed.","tokens_in":13763,"tokens_out":13862,"duration_ms":125771,"concrete_test":"Construct the one-dimensional counterexample above on U=(0,1): take u*(x)=-(x-1/2)^2, phi=0, samples {k/mr}_{k=0}^{mr}, and hmr(x)=u*(x)+mr^{-1} sin(2*pi*mr*x). Verify that (i) sup_mr [hmr]_{alpha;U_mr} <= [u*]_alpha < infinity, (ii) sup_{x in U_mr} |hmr(x)-u*(x)| = 0, yet (iii) for large mr, hmr-phi has no critical point at any sample point because hmr'(k/mr)=u*'(k/mr)+2*pi, which is nonzero. Therefore the asserted sequence x_mr in the proof of Theorem 4 does not exist and the viscosity argument fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4 shows uniform convergence only for the projected functions v_mr(x)=h_mr(p_mr(x)), not for the minimizers h_mr themselves. The functions v_mr are piecewise constant on the Voronoi cells of the training set (with jumps of size O(epsilon^alpha)), so they are not smooth and cannot be used in the viscosity test. The passage 'Following (22) there exists a sequence of points x_mr in T_mr^r union T_mr^b such that hmr-phi attains its local maximum at x_mr' is unjustified: (22) controls hmr-u* only on the training set, and condition (19) does not control hmr between samples. Explicitly, let U=(0,1), u*(x)=-(x-1/2)^2, phi=0, and put hmr(x)=u*(x)+A_mr sin(2*pi*mr*x) with A_mr=mr^{-1} and training set {k/mr}. On every sample hmr=u*, so (22) and sup_mr [hmr]_{alpha;U_mr}<infinity hold, but for large mr the derivative of hmr at every sample is dominated by the sine term and never vanishes, so hmr-phi has no local maximum at any sample point. Thus the viscosity-subsolution step cannot be carried out with the stated assumptions; the claimed convergence in C^0(closure U) is unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14099,"tokens_out":15271,"duration_ms":116683,"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":[{"comment":"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.","section":"Theorem 3 proof, line after Eq. (18)"},{"comment":"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.","section":"Theorem 4 proof, paragraph after Eq. (22)"},{"comment":"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.","section":"Theorem 4 proof, viscosity-subsolution paragraph"},{"comment":"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.","section":"Assumption 3"}],"minor_comments":[{"comment":"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.","section":"Theorem 3 proof, displayed limit"},{"comment":"The proof invokes 'Lemma B.1 in [31]' without stating it; please include the lemma or its statement to make the paper self-contained.","section":"Theorem 2 proof"},{"comment":"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.","section":"Theorem 4 proof"},{"comment":"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.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper is not ready for publication. The Theorem 4 gap is not a minor technicality: condition (19) is too weak, and the local-maximum inference is demonstrably false. Additionally, the regularization rate in Theorem 3 appears algebraically inconsistent. I recommend rejection rather than major revision, because repairing these issues would require a substantially different proof or significantly stronger assumptions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nTo 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.\n\nThe 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.\n\nRecommendation: 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.","headline":"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].","tokens_in":14564,"tokens_out":3381,"would_cite":false,"duration_ms":30450,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M12","68T07","41A46","35J25","35K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Physics Informed Neural Networks","Convergence","Viscosity Solutions","Fully nonlinear PDEs","Degenerate elliptic PDEs","PINN convergence","Hölder regularity","Differential Equations"],"falsifier":"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.","tokens_in":13594,"feed_emoji":"🤖","tokens_out":14698,"duration_ms":117473,"temperature":0.7,"pith_summary":"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.","feed_headline":"PINN minimizers converge to viscosity solutions of nonlinear PDEs","feed_subtitle":"Bounded Holder regularity on sampled points plus the comparison principle force almost-sure convergence to the unique solution.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the PINN convergence framework, the Hölder-regularized loss, the expected-versus-empirical loss comparison, and the space-filling Lemma B.2 that the proof adapts.","marker":"[31]"},{"why":"Provides the viscosity solution theory, the comparison principle, and Ishii's existence theorem used to identify and select the limit.","marker":"[8]"},{"why":"Defines the PINN loss and the empirical/expected loss setup that the paper minimizes.","marker":"[27]"},{"why":"Source of the probabilistic space-filling argument controlling distances from arbitrary points to the training samples.","marker":"[5]"},{"why":"Background for space-filling and Lipschitz-regularization estimates that support the sample-density bounds.","marker":"[14]"},{"why":"Supplies the variant of Arzelà–Ascoli used to extract a uniformly convergent subsequence of the projected functions.","marker":"[6]"},{"why":"Justifies use of global minimizers by ensuring epsilon-suboptimal global solutions exist in the network class.","marker":"[20]"}],"fun_headline_variants":["PINN minimizers converge to unique viscosity solutions","Hölder-regularized PINNs converge almost surely to viscosity solutions","Theorem: PINN minimizers converge to viscosity solutions","PINNs converge almost surely to viscosity solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["PINN minimizers converge to unique viscosity solutions","Hölder-regularized PINNs converge almost surely to viscosity solutions","Theorem: PINN minimizers converge to viscosity solutions","PINNs converge almost surely to viscosity solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001176,"raw_usage":{"total_tokens":4786,"prompt_tokens":799,"completion_tokens":3987,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":415,"completion_tokens_details":{"reasoning_tokens":3923}},"tokens_in":415,"tokens_out":3987,"duration_ms":21625,"temperature":1.0,"reasoning_tokens":3923,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:59:30.499975+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Communicat ions in Computational Physics 28(5), 2042–2074 (2020)","cited_arxiv_id":null,"evidence_quote":"Supplies the PINN convergence framework, the Hölder-regularized loss, the expected-versus-empirical loss comparison, and the space-filling Lemma B.2 that the proof adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the viscosity solution theory, the comparison principle, and Ishii's existence theorem used to identify and select the limit."},{"cited_title":"Journal of Com- putational Physics 378, 686–707 (2019)","cited_arxiv_id":null,"evidence_quote":"Defines the PINN loss and the empirical/expected loss setup that the paper minimizes."},{"cited_title":"SIAM Journal on Mathematics of Data Science 1(4), 780–812 (2019)","cited_arxiv_id":null,"evidence_quote":"Source of the probabilistic space-filling argument controlling distances from arbitrary points to the training samples."},{"cited_title":"arXiv preprint arXiv:1808.0954 0 (2018)","cited_arxiv_id":null,"evidence_quote":"Background for space-filling and Lipschitz-regularization estimates that support the sample-density bounds."},{"cited_title":"SIAM Journal on Numerical Analysis 53(1), 82–104 (2015)","cited_arxiv_id":null,"evidence_quote":"Supplies the variant of Arzelà–Ascoli used to extract a uniformly convergent subsequence of the projected functions."},{"cited_title":"Mathematical programming 173(1-2), 221– 249 (2019)","cited_arxiv_id":null,"evidence_quote":"Justifies use of global minimizers by ensuring epsilon-suboptimal global solutions exist in the network class."}],"review_version":1}