{"id":"4a1869a5-e55a-4ec9-ac7a-0d3c2226f6ad","arxiv_id":"2505.07311","paper_version":1,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Projected gradient descent for one-hidden-layer PINNs solving the Poisson equation is claimed to converge with average loss O(1/sqrt(T) + 1/sqrt(m)) plus an approximation error, with a generalization bound via Rademacher complexity of the Laplacian.","lead":"This paper gives a theoretical bound on how fast projected gradient descent reduces the training loss for physics-informed neural networks solving the Poisson equation. It claims an O(1/sqrt(T) + 1/sqrt(m)) error rate without requiring very wide networks, a regime closer to how PINNs are actually trained.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4's uniform linearization bound is false: the projection radius p√m gives per-neuron Laplacian remainders of order p², not O(p²/√m), so the Lyapunov step in Theorem 6 is unsupported.","rationale":"The reader's weakest assumption identifies exactly the scaling of Lemma 4, and my reading confirms that this is the most load-bearing point in the proof of Theorem 6. A concrete adversarial evaluation of Eq. (19) shows the lemma is false as stated: under the projection radius p√m, the second-order remainder of the Laplacian map is not O(1/√m) but can be order p² per active neuron, and the proof's line marking (θ_ij−θ_ij(0))² as O(1/m) is algebraically wrong inside the stated ball. Since the lower-bound step of the Lyapunov argument relies on this uniform linearization error to replace the linearized inner product with the empirical loss, Theorem 6 does not follow from the presented proof. The additional O(m p²/√T) scaling of the initial potential reinforces the concern: even a corrected Lemma 4 would not yield the m-independent C1/√T rate claimed in Eq. (24). The experimental section does not test this m-dependence because it fixes T=m and uses only a few small widths, so it cannot validate the asymptotic scaling. These are internal inconsistencies in the argument, not disagreements with scientific consensus, and they are severe enough to support the reader's REJECT verdict; the underlying strategy may be salvageable, but the current proof does not establish the theorem.","tokens_in":20270,"tokens_out":18969,"duration_ms":190960,"concrete_test":"Evaluate Eq. (19) analytically at m=10^4, d=1, p=1, σ(t)=sin(t), θ(0)=0, θ_1=√m, θ_i=0 for i>1, and x=1. The left side is |sin(√m)|, and the claimed right side is C/√m; if |sin(√m)|·√m is unbounded, Lemma 4 is refuted. Re-run the same calculation with θ(0)=ε=10^{-3} to confirm the failure persists on a positive-probability set under the truncated Gaussian initialization. As an additional check, re-derive the second-order Taylor remainder for a single neuron and verify that sup_{∥δ_i∥≤p√m} |φ_i(θ_i0+δ_i)−φ_i(θ_i0)−Dφ_i(θ_i0)δ_i| is at least p² for this adversarial choice, independent of m.","verdict_should_be":"REJECT","load_bearing_attack":"Lemma 4 (Eq. 19) asserts that for every θ_i∈B2(θ_i(0), p√m) the Laplacian residual after first-order expansion is O(p²/√m). This is not true under the stated radius. With d=1, σ(t)=sin(t), θ(0)=0, x=1, and θ_1=p√m (all other θ_i=θ_i(0)), the initial Laplacian and its θ-gradient vanish, so the left side of Eq. (19) equals p²|sin(p√m)|, while the RHS is C p²/√m; for large m the inequality fails, and by continuity it fails on a positive-probability neighborhood of this initialization. The appendix proof marks (θ_ij−θ_ij(0))² ≾ 1/m, but inside the projection ball this squared displacement can be O(p²m). This uniform linearization bound is precisely the mechanism that turns the gradient inner product into 2E_St(θ(t)) in the lower-bound part of the proof of Theorem 6 ('Lemmas 4, 5 and 12'), so the Lyapunov drift inequality and the claimed 1/√T + 1/√m rate do not follow. Separately, the initial-potential term L(θ(0))/(4ηT) is O(m p²/√T), so even with a repaired Lemma 4 the theorem's m-independent C1/√T term in Eq. (24) is not justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes projected gradient descent for two-layer physics-informed neural networks solving the Poisson equation. It claims a high-probability, non-asymptotic bound on the time-averaged empirical PINN loss of the form O(1/sqrt(T) + 1/sqrt(m) + epsilon_p), where epsilon_p is the approximation error of the true solution by functions in a neural-tangent-kernel RKHS class with bounded transport map. The proof strategy combines a Lyapunov drift argument, a uniform linearization error bound under a projection ball, approximation estimates for the linearized network, and Rademacher-complexity bounds for the network and its Laplacian. These results are then combined with a regularity-theory estimate to obtain an overall H^{1/2} error bound. The paper also reports a small numerical experiment on a two-dimensional Poisson problem.","tokens_in":20603,"tokens_out":13556,"duration_ms":130220,"significance":"If the main theorem were correct, the paper would make a significant contribution: it would provide one of the first non-over-parameterized optimization guarantees for PINNs, with explicit rates and without fitted constants. The use of an externally defined approximation error epsilon_p, rather than a circularly fitted quantity, is a genuine strength, as is the attempt to treat freshly sampled quadrature points. The Rademacher-complexity analysis for the Laplacian of the network is also a natural and potentially useful ingredient. However, the central proof contains multiple scaling errors, including a false uniform linearization bound, and these errors affect the main theorem, the generalization proposition, and the overall error theorem.","major_comments":[{"comment":"The uniform linearization bound in Eq. (19) is false under the stated projection radius. Since theta_i is only constrained to B_2(theta_i(0), p*sqrt(m)), the squared displacement |theta_ij - theta_ij(0)|^2 can be O(p^2 m), not O(1/m) as the proof marks in the appendix. Concrete counterexample: take d=1, sigma(t)=sin(t), theta_i(0)=0 for all i, and symmetric output weights c_i = +/-1/sqrt(m). Move only one neuron by setting theta_1 = p*sqrt(m), leaving its symmetric partner at 0. Then F(.;theta(0))=0 and Delta F(.;theta(0))=0, and the theta-gradient of Delta F at theta(0) is also zero, so the left-hand side of Eq. (19) at x=1 equals p^2 sqrt(m) |sin(p sqrt(m))|, while the right-hand side is C p^2/sqrt(m). For large m the inequality fails. The appendix proof's step '(theta_ij - theta_ij(0))^2 ≾ 1/m' is exactly where the argument breaks. Since Lemma 4 is the mechanism that converts the gradient inner product into 2 E_{S_t}(theta(t)) in the lower-bound part of the proof of Theorem 6, the Lyapunov drift inequality and the claimed rates do not follow.","section":"Section 3.2, Lemma 4 (Eq. (19))"},{"comment":"The upper bound on the quantity (*) loses a factor of m. The per-neuron bound just before Eq. (27) gives ||grad_{theta_i} E_{S_t}(theta(t))||_2 <= c_4 + c_5 sqrt(m). The quantity (*) is the sum over i=1,...,m of the squares of these norms, so it is at most m(c_4 + c_5 sqrt(m))^2, not (c_4 + c_5)^2 as written in Eq. (28). This is not a cosmetic issue: after summation with eta = 1/sqrt(T), the term (1/4) eta^2 (*) would contribute O(m^2/sqrt(T)), which destroys the claimed C_1/sqrt(T) term independent of m.","section":"Section 3.3, proof of Theorem 6, Eqs. (27)-(28)"},{"comment":"Lemma 12 is internally inconsistent: the displayed computation for Eq. (33) ends with '<= p^2 C_Omega^2 sigma_2 sqrt(m)', which contradicts the lemma's claim that the quantity is ≾ 1/sqrt(m). Each of the m summands inside the outer sum is O(p^2/sqrt(m)) after the intermediate bounds, so the total is O(p^2 sqrt(m)) for fixed p. This estimate is used in the lower-bound step of Theorem 6 to control the difference between the gradient inner product at theta(t) and at theta(0); with an O(sqrt(m)) remainder, the conversion into 2 E_{S_t}(theta(t)) up to O(1/sqrt(m)) terms is not valid. The second estimate (34) is asserted with a bare appeal to Lemma 14 and has the same scaling risk.","section":"Appendix, Lemma 12, Eq. (33)"},{"comment":"The claimed 1/sqrt(T) term is also not justified by the initial-potential term. With the reference weight theta chosen by Eq. (18), ||theta_i(0) - theta_i||_2 = ||v(theta_i(0))||_2 <= p, so L(theta(0)) = sum_i ||theta_i(0)-theta_i||_2^2 <= m p^2. Since eta = 1/sqrt(T), the term L(theta(0))/(4 eta T) is O(m p^2/sqrt(T)), not O(1/sqrt(T)) with a constant independent of m. The final displayed inequality introduces an undefined symbol 'alpha^2' for this term, and no m-independent bound is supplied.","section":"Section 3.3, final display of the proof of Theorem 6"}],"minor_comments":[{"comment":"The lemma states that the bounds hold 'for all x in Omega' with probability at least 1-delta, but the proof only applies Hoeffding's inequality pointwise in x. Some uniform covering or continuity argument is needed to justify the simultaneous statement over the uncountable set Omega.","section":"Section 3.2, Lemma 5"},{"comment":"The sentence 'In the proof of Theorem 6 we have shown that ||grad_{theta_i} Delta F(x;theta(t))||_2 ≾ 1/sqrt(m)' is contradicted by the actual calculation in the proof of Theorem 6, which gives O(m + sqrt(m)) for general theta(t) in the projection ball. If the Rademacher argument only needs this gradient at theta(0), the statement should say so explicitly.","section":"Section 4, proof of Proposition 9"},{"comment":"The experimental section does not state the truncation parameter a used for the initialization distribution, nor the precise initialization of the output weights, so the reported numerical results are not fully reproducible from the text.","section":"Section 3.4, numerical example"}],"recommendation":"reject","confidential_remarks":"The paper has several attractive ideas, especially the fresh-sample setting and the use of an external approximation error, but the central proof is not sound. The false uniform linearization bound in Lemma 4 is not a local fix: the projection radius p*sqrt(m) is the source of the problem, and changing it would require reworking the algorithm, the RKHS class definition, and the Lyapunov argument. Additional independent m-scaling errors in the gradient-sum bound and Lemma 12 reinforce the conclusion that the main theorems are not established in the submitted form. I recommend rejection, while noting that a substantially revised version with a corrected proof might be worth reconsidering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the first paper I know of to target a non-asymptotic optimization guarantee for PINNs without quadratic over-parameterization and with fresh collocation points drawn each iteration. That is a genuine gap, and the problem choice is right. Second, the central estimate does not close, and the stress-test note is correct: Lemma 4 is false at the stated projection radius.\n\nThe proof machinery is Ji-Telgarsky's Lyapunov drift, honestly cited, and the actual new pieces are the Laplacian in the loss and a Rademacher complexity bound for the class of network Laplacians. The paper is also candid: Remark 1 admits the projection is artificial and that choosing p is nontrivial; Remark 3 states the activation restrictions plainly; and the toy experiment is modest, reporting the 90% percentile of the time-averaged loss without overclaiming. The citations look fair, and the use of the authors' earlier regularity estimate [35] is legitimate, not circular.\n\nThe flaws are load-bearing though. Lemma 4 claims that inside the projection ball of radius p sqrt(m), the Laplacian linearization error is O(p^2/sqrt(m)) uniformly over x. But one neuron at the ball's boundary can change the Laplacian by order p^2 sqrt(m). Take sigma(t)=sin(t), d=1, theta(0)=0, and one hidden weight at p sqrt(m): the linearization is identically zero at that initialization, while the exact Laplacian is (1/sqrt(m)) c_1 p^2 m sin(p sqrt(m) x). The failure is a factor of m, and it persists on a positive-probability set of initializations by continuity. Theorem 6's proof uses Lemma 4 exactly to lower-bound the gradient inner product by 2 E_{S_t}, so the drift inequality and the stated rate do not follow.\n\nThe m-losses continue downstream. The initial potential L(theta(0))/(4 eta T) is of order m p^2 / sqrt(T), so the claimed m-independent C_1/sqrt(T) term in Eq. (24) is unsupported. The gradient-norm bound (27) is per-neuron and grows with m (or worse), so summing over m neurons leaves extra powers of m that disappear from the final inequality. And Proposition 9's proof leans on a 1/sqrt(m) bound for a gradient quantity that Theorem 6's proof bounds by an m-growing expression; as written, the Rademacher bound is not supported by the estimate it cites.\n\nThe errors all push toward optimistic rates. A repair would need a smaller projection radius, a different Lyapunov potential, or genuinely uniform second-order control. As written, the paper should not be accepted, but the failure is specific and the enterprise is worth a revision. This is a paper for the PINN-theory and NTK-theory crowd; a competent referee can write a constructive, checkable report. Send it to referees rather than desk-rejecting.","headline":"Right question, coherent strategy - but Lemma 4's linearization bound is off by a factor of m, so Theorem 6 is not established as written; the paper deserves referees and a serious revision.","tokens_in":21094,"tokens_out":17019,"would_cite":false,"duration_ms":149482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65N12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a non-asymptotic convergence guarantee for projected gradient descent on physics-informed neural networks solving the Poisson equation, at a rate $O(1/\\sqrt T + 1/\\sqrt m + \\epsilon_p)$, without over-parameterization.","keywords":["physics-informed neural networks","neural tangent kernel","reproducing kernel Hilbert space","projected gradient descent","Poisson equation","non-asymptotic convergence","Rademacher complexity","under-parameterized regime"],"falsifier":"Take a smooth activation satisfying Assumption 2, set $x=(1,\\ldots,1)$, and put every weight on the boundary of its projection ball, $\\theta_i=\\theta_i(0)+p\\sqrt m\\,e_1$ for all $i$. For a sequence of widths $m=10^2,10^3,\\ldots,10^6$, compute the normalized remainders $R_m=\\sqrt m\\,|\\Delta F(x;\\theta)-\\sum_i\\nabla_{\\theta_i}\\Delta F(x;\\theta(0))^T(\\theta_i-\\theta_i(0))|$ and the analogous $F$-remainder. Lemma 4 predicts $R_m$ stays bounded uniformly in $m$ and in the direction of $\\theta-\\theta(0)$; a naive second-order Taylor estimate grows with $m$. If $R_m$ diverges, the linearization step in the proof of Theorem 6 is unsupported.","tokens_in":20048,"feed_emoji":"📉","tokens_out":15585,"duration_ms":141190,"temperature":0.7,"pith_summary":"This paper proves a finite-time convergence guarantee for projected gradient descent on a one-hidden-layer physics-informed neural network (PINN) trained to solve the Poisson equation. The main theorem bounds the time-averaged empirical loss by $O(1/\\sqrt T + 1/\\sqrt m + \\epsilon_p)$ with high probability over the random initialization, where $T$ is the number of gradient steps, $m$ is the network width, and $\\epsilon_p$ is the error of approximating the true solution by functions in a neural-tangent-kernel space with bounded transport map. The result does not require the network width to be much larger than the number of collocation points, and it permits fresh integration points at every iteration — the setting in which PINNs are usually trained. A companion generalization bound, built from Rademacher complexities of the network and its Laplacian, converts the loss bound into an overall $H^{1/2}$ error estimate. The authors present this as the first non-over-parameterized optimization guarantee for PINNs.","feed_headline":"No wide networks needed: PINN training provably converges","feed_subtitle":"Fresh collocation points and any width are covered by a new 1/sqrt(T) + 1/sqrt(m) loss bound.","key_machinery":"The argument is carried by the Lyapunov potential $L(\\theta)=\\sum_{i=1}^m\\|\\theta_i-\\bar\\theta_i\\|_2^2$, where $\\bar\\theta$ reconstructs an oracle approximant in $F_p$ from its transport map; nonexpansivity of the projection gives the drift inequality. Two bounds feed into it. Lemma 4 (the linearization error) states that inside the projection ball, the network and its Laplacian differ from their first-order Taylor expansions at initialization by at most $C\\|x\\|_2^2p^2/\\sqrt m$, uniformly in $\\theta$. Lemmas 5 and 12 use the random features $\\Phi_{\\mathrm{NTK}}(x;\\theta)=x\\sigma'(\\theta^T x)$ to show that the linearized model approximates $\\Delta u^*$ and the boundary trace of $u^*$ up to $O(\\sqrt{\\log(1/\\delta)/m})$ plus the approximation error $\\epsilon_p$. The symmetric initialization makes $F(x;\\theta(0))=\\Delta F(x;\\theta(0))=0$, so the linear model starts at the zero function, and the projection keeps the gradient norm bounded while making the linearization error uniform.","core_discovery":"At the center of the paper is Theorem 6. For a network $F(x;\\theta)=m^{-1/2}\\sum_{i=1}^m c_i\\sigma(\\theta_i^T x)$ with a smooth activation whose first four derivatives are bounded, symmetric initialization, and projection onto balls of radius $p\\sqrt m$ around the initial weights, projected gradient descent with step size $\\eta=1/\\sqrt T$ satisfies, for any choice of integration sets $S_t$, $$\\frac{1}{T}\\sum_{t=0}^{T-1}E_{S_t}(\\$\\theta$(t)) \\le \\frac{C_1}{\\sqrt T}+C_2\\sqrt{\\frac{\\log(\\max\\{b_\\$\\Omega$,b_{\\partial\\$\\Omega$}\\}/\\delta)}{m}}+C_3\\epsilon_p$$ with probability at least $1-2\\delta$ over the random initialization. Here $\\epsilon_p$ is the best error with which $u^*$ can be represented by functions in the class $F_p$, the neural-tangent-RKHS ball of radius $p$. The paper then proves a Rademacher-complexity bound for the network and its Laplacian, giving a generalization bound for the best empirical-loss iterate, and combines both with the regularity estimate $\\|u^*-F\\|_{H^{1/2}}\\lesssim\\sqrt{E(\\theta)}$ to obtain the overall error in Theorem 8. The intended contribution is to move PINN optimization theory from the regime of quadratic over-parameterization to the regime of arbitrary width and online sampling.","pith_inferences":["I read the projection as the actual mechanism of the result: it confines the trajectory to the region where the uniform linearization bound holds. A testable consequence is that unprojected gradient descent should lose the promised rate whenever the weights leave the ball of radius $p\\sqrt m$; comparing projected and unprojected runs with identical initialization and step size would show this.","The proof structure is generic in the PDE: the Poisson equation enters only through the Laplacian in the residual and a boundary trace term, and Assumption 2 only controls four derivatives of the activation. I would expect the same Lyapunov argument to carry over to other linear elliptic equations with the same second-order structure, with $\\epsilon_p$ redefined for the corresponding differential ","An empirical stress test of Lemma 4 would be to compute the normalized linearization remainder over the boundary of the projection ball for growing width. If it is not uniform in the direction of $\\theta-\\theta(0)$, the convergence claim might still hold, but the paper's proof would need a different linearization estimate.","The output weights are frozen at their Rademacher initialization; training both layers would require a different potential and a different function class, since $F_p$ is defined through hidden-weight transport maps only. Extending the guarantee to joint training is a natural next step not covered here."],"forward_implications":["Because Theorem 6 holds for arbitrary integration sets $S_t$, a practitioner may re-sample collocation points at every gradient step and still have the same $1/\\sqrt T + 1/\\sqrt m$ guarantee; the proof does not need a fixed dataset.","The absence of over-parameterization means the width $m$ can be smaller than the number of collocation points — the regime reached as soon as the sampling density exceeds the network width — with the rate degrading only as $1/\\sqrt m$.","When the solution $u^*$ is realizable, so $u^*\\in F_\\alpha$ for some $\\alpha$, choosing $p\\ge\\alpha$ makes the approximation term $\\epsilon_p$ vanish and yields an unconditional $O(1/\\sqrt T + 1/\\sqrt m)$ empirical-loss rate; the constants grow polynomially with $p$.","With iid uniform collocation points, the best-iterate generalization error is bounded by $C_4/\\sqrt m + C_5\\sqrt{\\log(1/\\delta)/\\min\\{b_\\Omega,b_{\\partial\\Omega}\\}}$, and combining this with Theorem 6 and the regularity estimate gives the overall $H^{1/2}$ error bound of Theorem 8."],"supporting_citations":[{"why":"Supplies the Lyapunov drift technique that removes over-parameterization in the supervised setting and is adapted here to the PINN loss.","marker":"[25]"},{"why":"Defines the neural tangent kernel and linearized feature dynamics used to construct the class $F_p$ and the approximation error.","marker":"[24]"},{"why":"Provides the over-parameterized exponential convergence result for two-layer PINNs that Theorem 6 is the non-over-parameterized counterpart to.","marker":"[31]"},{"why":"Second over-parameterized PINN convergence baseline whose quadratic-width assumption is dropped in this paper.","marker":"[21]"},{"why":"Provides the regularity-theory estimate $\\|u^*-F(\\cdot;\\theta)\\|_{H^{1/2}}\\le c_{\\mathrm{reg}}\\sqrt{E(\\theta)}$ used to turn loss bounds into PDE error bounds in Theorem 8.","marker":"[35]"},{"why":"Gives the Rademacher-complexity bound for the linearized network class used in Lemma 13.","marker":"[4]"},{"why":"Supplies the Rademacher-complexity risk bounds used in Proposition 10 for the generalization guarantee.","marker":"[5]"}],"fun_headline_variants":["Arbitrary width PINNs: optimization converges","No over-parameterization needed for PINN convergence","Projected GD for PINNs: rate O(1/sqrt(T))","Online sampling: PINN loss bound for any width","PINN training guarantee without width scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the uniform linearization bound in Lemma 4: inside the projection ball of radius $p\\sqrt m$, both the network and its Laplacian stay within $O(1/\\sqrt m)$ of their first-order Taylor expansions at initialization, and if that bound is false, the drift inequality that produces Theorem 6 has no foundation.","fun_headline_variants_meta":{"raw":{"variants":["Arbitrary width PINNs: optimization converges","No over-parameterization needed for PINN convergence","Projected GD for PINNs: rate O(1/sqrt(T))","Online sampling: PINN loss bound for any width","PINN training guarantee without width scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1493,"prompt_tokens":1017,"completion_tokens":476,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":399}},"tokens_in":633,"tokens_out":476,"duration_ms":4611,"temperature":1.0,"reasoning_tokens":399,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:23:32.050925+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth activation satisfying Assumption 2, set $x=(1,\\ldots,1)$, and put every weight on the boundary of its projection ball, $\\theta_i=\\theta_i(0)+p\\sqrt m\\,e_1$ for all $i$. For a sequence of widths $m=10^2,10^3,\\ldots,10^6$, compute the normalized remainders $R_m=\\sqrt m\\,|\\Delta F(x;\\theta)-\\sum_i\\nabla_{\\theta_i}\\Delta F(x;\\theta(0))^T(\\theta_i-\\theta_i(0))|$ and the analogous $F$-remainder. Lemma 4 predicts $R_m$ stays bounded uniformly in $m$ and in the direction of $\\theta-\\theta(0)$; a naive second-order Taylor estimate grows with $m$. If $R_m$ diverges, the linearization step in the proof of Theorem 6 is unsupported.","supporting_citations":[{"cited_title":"Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks","cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov drift technique that removes over-parameterization in the supervised setting and is adapted here to the PINN loss."},{"cited_title":"Neural tangent kernel: Convergence and generalization in neural networks","cited_arxiv_id":null,"evidence_quote":"Defines the neural tangent kernel and linearized feature dynamics used to construct the class $F_p$ and the approximation error."},{"cited_title":"Chapter 11 - Two-layer neural networks for partial differential equations: optimization and generalization theory","cited_arxiv_id":null,"evidence_quote":"Provides the over-parameterized exponential convergence result for two-layer PINNs that Theorem 6 is the non-over-parameterized counterpart to."},{"cited_title":"Gradient descent finds the global optima of two- layerphysics-informedneuralnetworks","cited_arxiv_id":null,"evidence_quote":"Second over-parameterized PINN convergence baseline whose quadratic-width assumption is dropped in this paper."},{"cited_title":"Notes on Exact Boundary Values in Residual Minimisation","cited_arxiv_id":null,"evidence_quote":"Provides the regularity-theory estimate $\\|u^*-F(\\cdot;\\theta)\\|_{H^{1/2}}\\le c_{\\mathrm{reg}}\\sqrt{E(\\theta)}$ used to turn loss bounds into PDE error bounds in Theorem 8."}],"review_version":1}