{"id":"ea7583d4-6445-4088-878e-b98e139f0d1f","arxiv_id":"2411.14151","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A priori error estimates for two-layer ReQU/ReCU networks solving 2n-order elliptic equations with non-homogeneous Dirichlet, Neumann, and Robin boundary conditions.","lead":"This paper proves error bounds for the Deep Mixed Residual Method when it solves high-order elliptic equations with neural networks. The bounds show the method can avoid the curse of dimensionality, but only under strong smoothness assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.2 is not supported by the stated lemmas: Lemma 3.7 proves only an H^1 approximation rate, while Eq. (38) and Lemma 3.5 require H^2 approximation for the second-order MIM error.","rationale":"The paper's central claim is that MIM yields dimension-exempt a priori error bounds for high-order elliptic equations under both first-order and second-order least-squares formulations. The first-order part (Theorem 2.1) is largely supported by Lemma 3.6, whose H¹ approximation statement matches the H¹/H(div) approximation error Eapp, modulo the Dirichlet square-root scaling issue already noted by the reader. My concern targets the second-order part: Theorem 2.2, Lemma 3.5, and Eq. (38) require H² approximation, but Lemma 3.7 and the entire Appendix A.1 establish only H¹ approximation. This is a concrete proof gap, not merely a strong regularity assumption. The second-order system is a principal advertised contribution, so the omission is load-bearing. The reader's weaker-assumption identification (Barron regularity) is legitimate, but it is an external regularity assumption rather than an internal mismatch in the proof; I therefore only partially agree. The proof gap may be repairable: ReCU networks are piecewise cubic and finite-difference constructions could plausibly be extended to H², but the paper does not provide that argument. Since the substantial coercivity analysis and the first-order theorem remain, a conditional verdict is still appropriate, and my read does not change the reader's verdict.","tokens_in":31829,"tokens_out":20044,"duration_ms":204181,"concrete_test":"Re-derive Lemma A.2 (Appendix A.1) with the H² norm: starting from the ReQU construction (89) and the identity (90), check whether the finite-difference coefficients yield ||g − g_m||²_{H²([−1,1])} ≤ C B²/m. If no such bound follows, then Eq. (38)'s E*_app is not controlled by Lemma 3.7, and Theorem 2.2 lacks the required approximation estimate. As a numerical cross-check, apply the Appendix A.1 construction to u*(x)=cos(π x_1), which lies in every B^s, and compute ||φ_0^{(m)} − u*||²_{H²(Ω)} for m = 8, 16, 32, 64; if this quantity does not decay at least as fast as C/m, the missing H² approximation is not automatically supplied by the existing construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The second-order theorem is not derivable from the stated approximation estimate. In Section 3.2.2, the proof applies Céa's lemma with the approximation error E*_app = inf_{vθ} Σ_{k=0}^{n-1} ||φ_k − Δ^k u*||²_{H²(Ω)} (Eq. 38), because the bilinear form Bα(v,w) is bounded on (H²)^n (Lemma 3.2) and the Neumann/Robin traces require ∂nφ_k. However, Lemma 3.7 and its proof in Appendix A.1 (Lemma A.2 and identity (91)) only construct ReCU networks with an H¹ approximation bound: ||u* − u_m||²_{H¹} ≼ ||u*||²_{B^{2n+3}}/m. No estimate is given for ||Δ(φ_k − Δ^k u*)||²_{L²} or for normal-derivative traces. Since H¹ control does not imply H² control, the Céa step in Lemma 3.5 cannot deliver the claimed H(Ω)-type error in Theorem 2.2. This is an internal gap, independent of whether the Barron regularity assumption holds. A repair would require a new H² approximation lemma for ReCU networks, or a reformulation of the loss/coercivity in weaker norms.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes the Deep Mixed Residual Method (MIM) for 2n-order elliptic equations with non-homogeneous Dirichlet, Neumann, and Robin boundary conditions. Two least-squares formulations are studied: a first-order system approximated by two-layer ReQU networks and a second-order system approximated by two-layer ReCU networks. Using spectral Barron spaces, a Céa-lemma framework from [32], perturbation-based coercivity estimates, and Rademacher complexity bounds, the authors claim a priori error estimates of the form O(||u*||^2/m + ||u*||^2/N^{1/2}) for Neumann/Robin and O(||u*||^2/m^{1/2} + ||u*||^2/N^{1/4}) for Dirichlet, with constants depending at most polynomially on dimension. The proof consists of coercivity lemmas, approximation lemmas for Barron functions, generalization bounds, and an error decomposition; the optimization error is explicitly left out of scope.","tokens_in":32115,"tokens_out":19101,"duration_ms":196407,"significance":"If the results were fully supported, the paper would make a valuable contribution: it extends the MIM error analysis to high-order equations with non-homogeneous boundary conditions, gives dimension-exempt rates in terms of network width m and sample sizes N, and identifies that MIM lowers the activation-function regularity relative to the deep Ritz method. The perturbation argument with the carefully chosen parameter sequence delta_k and the resulting coercivity estimates for the coupled first-order and second-order systems are technically substantial. The treatment of non-homogeneous Neumann and Robin conditions, including zero-mean penalties, is also a strength. However, the second-order theorem as written is not derivable from the stated lemmas, and the Dirichlet sup-linear coercivity step needs clarification; these are load-bearing gaps that prevent the paper from being accepted in its current form.","major_comments":[{"comment":"Theorem 2.2 is not supported by the stated approximation estimate. The Céa step in Lemma 3.5 requires E*_app = inf over FReCU,m of sum_k ||phi_k - Delta^k u*||^2_{H^2(Omega)}, because the second-order bilinear form is bounded on (H^2)^n and the Neumann/Robin traces involve normal derivatives. However, Lemma 3.7 only establishes an H^1 approximation rate, and its proof in Appendix A.1 (Lemma A.2 and the construction around Eq. (92)) yields ||g - g_m||^2_{H^1} <= C B^2/m with no control of Delta(g - g_m) in L^2. Since H^1 closeness does not imply H^2 closeness, the bound on E*_app does not follow, and the H(Omega)-type error estimate in Theorem 2.2 cannot be concluded. A repair requires either a new H^2 approximation lemma for ReCU networks or a reformulation of the second-order loss/coercivity in weaker norms.","section":"Section 3.3, Eq. (38), Lemma 3.7, and Appendix A.1"},{"comment":"The Dirichlet sup-linear coercivity statement needs correction or clarification. The proof of Lemma 4.1 establishes the homogeneous inequality (B_D^{1/2}(u,u) + sum_k ||psi_k||_{L^2(dOmega)}) B_D^{1/2}(u,u) >= C sum_k (||phi_k||^2_{H^1} + ||psi_k||^2_{H(div)}). When ||u||_{H^1} <= E is imposed, the trace and boundedness estimates give B_D^{1/2}(u,u) >= (c/E) sum_k (||phi_k||^2_{H^1} + ||psi_k||^2_{H(div)}), not c E times that sum as displayed. The factor E^{-1} is essential: it must be propagated through Céa's lemma to obtain the scaling ||u*||^2/m^{1/2} in the Dirichlet parts of Theorems 2.1 and 2.2. Please correct the statement of Lemma 3.1 and indicate explicitly how E enters the constants in Lemma 3.3 and the final estimates.","section":"Section 3.1.1, Lemma 3.1, and Section 4.4, Eq. (48)"},{"comment":"The generalization bound for the second-order Neumann and Robin cases is incomplete. For alpha = N, R, the boundary loss classes S*_alpha contain normal derivatives partial_n phi_k. The proof of Lemma 3.10 estimates R_hatN(S*_alpha) using only R_hatN(FReCU,m) and a supremum bound on ||S*_alpha v_theta - g_alpha||_Linfty, but partial_n v_theta is not an element of FReCU,m, so Lemma A.5 does not apply to it. A separate Rademacher bound for the derivative network class {partial_n v_theta : v_theta in FReCU,m} is needed to justify the N^{-1/2} boundary contribution in Theorem 2.2 for Neumann and Robin conditions.","section":"Section 3.4, Lemma 3.10, and Appendix A.2"}],"minor_comments":[{"comment":"The statement contains a typo: the error bound is written with ||phi*_alpha||^2_{B^{2n+3}} but should refer to ||u*_alpha||^2_{B^{2n+3}}.","section":"Theorem 2.2"},{"comment":"The text says 'by applying Lemma 3.1 and Céa Lemma' in the second-order case; the correct reference is Lemma 3.2.","section":"Section 3.2.2"},{"comment":"The original loss in Eq. (2) displays the terms ||phi_{i+1} - div psi_i||^2 with an L^2(dOmega) boundary norm, whereas the analyzed first-order loss in Eq. (14) uses the interior L^2(Omega) norm through the operator P. Please clarify the discrepancy.","section":"Eq. (2) versus Eq. (14)"},{"comment":"The Dirichlet rate is printed as ||u*||^2/(4\\sqrt{N}), which is easy to misread as a fourth root only from the text; use \\sqrt[4]{N} consistently.","section":"Theorem 2.1 and Theorem 2.2"},{"comment":"The paper assumes u* belongs to B^{2n+2} or B^{2n+3} and asserts that elliptic regularity theory can guarantee this for smooth data, but no theorem or argument is supplied. Since the m^{-1} approximation rate and the dimension-exemption conclusion fail if u* is not in the relevant Barron space, this assumption should be stated explicitly as a regularity condition rather than as a consequence of standard elliptic regularity.","section":"After Eq. (9)"},{"comment":"The network classes are defined using the unknown norm ||u*|| as an a priori bound on the parameters, making the approximation and generalization statements target-aware. This is acceptable as an a priori estimate, but it should be stated that the admissible set is not known in practice.","section":"Eqs. (10) and (18)"},{"comment":"For the Neumann problem, the exact solution is unique only up to a constant; the paper should specify that u*_N denotes the unique zero-mean solution, so that the zero-mean penalty in the loss does not change the minimizer.","section":"Section 2.3.1, Neumann loss"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the missing H^2 approximation for ReCU networks in the second-order theorem; this is an internal proof gap rather than a disagreement with the literature. The first-order theorem and the coercivity machinery appear plausible and worth salvaging. If the authors can supply an H^2 approximation lemma or reformulate the second-order analysis in a weaker norm so that the H^1 approximation suffices, the paper could become acceptable. The Dirichlet sup-linear coercivity scaling issue also needs a clear correction, though it is likely fixable by tracking the E^{-1} factor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: the first-order part of this paper is a solid contribution; the second-order theorem is not supported by the proof as written. The gap is specific and likely repairable, not a reason to ignore the paper.\n\nWhat's new and good: they give the first a priori, dimension-exempt error estimate for a mixed neural method on high-order elliptic equations with non-homogeneous boundary data. The perturbation-based coercivity for high-order systems is a real technical step, and the non-homogeneous Dirichlet/Neumann/Robin treatment goes beyond the homogeneous cases in the DRM/PINN literature. The rates for Neumann/Robin (1/m + 1/sqrt(N)) are plausible, and the Dirichlet rates (1/sqrt(m) + 1/N^{1/4}) follow from the sup-linear coercivity in a standard way—the reader's worry about the square-root step doesn't hold up; that step matches the theorem.\n\nThe soft spots, in proportion:\n\n1. The second-order theorem (Thm 2.2) is the real problem. The Céa argument in Lemma 3.5 uses an approximation error in H^2 (Eq. 38), because the bilinear form is bounded on H^2. But Lemma 3.7—whose proof is in the appendix—only delivers H^1 approximation for the ReCU networks. H^1 control does not give H^2 control, so the claimed Neumann/Robin rates for the second-order system do not follow. A repair either needs an H^2 approximation lemma for ReCU, or a reformulation of the loss/coercivity in weaker norms. This is internal and independent of the Barron assumption.\n\n2. The Barron regularity assumption (u* in B^{2n+2} or B^{2n+3}) is asserted rather than proven. Smoothness alone doesn't guarantee Barron membership; the spectral decay is a separate condition. The network classes are also defined using the unknown solution norm, which makes the approximation target-aware. These are assumptions a reader should carry.\n\n3. Minor: the sentence before Lemma 3.4 swaps the coercivity cases (says Dirichlet/Robin are coercive and Neumann is sup-linear; the opposite is true and the lemma itself is correct).\n\nWho this is for: anyone working on neural network analysis for PDEs, especially the MIM/DRM/PINN theory crowd. It deserves a serious referee, because the first-order result is likely correct and the second-order gap is specific and fixable. I'd want to see the revised version before citing the second-order claims.\n\nRecommendation: send to review, but flag the H^2 gap clearly.","headline":"First-order MIM bounds are credible and new; the second-order theorem has a real but repairable gap—an H^1 approximation lemma where H^2 is required.","tokens_in":32679,"tokens_out":4800,"would_cite":true,"duration_ms":42416,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N15","68Q25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a priori error bounds for the deep mixed residual method showing that two-layer ReQU and ReCU networks solve high-order elliptic equations at dimension-exempt rates, provided the exact solution lies in a spectral Barron…","keywords":["deep mixed residual method","high-order elliptic equations","a priori error analysis","Barron space","Rademacher complexity","coercivity","curse of dimensionality","least squares systems"],"falsifier":"Construct a sequence of exact solutions of the elliptic equation whose Barron norm grows with the dimension but whose classical Sobolev regularity is fixed, and run MIM with ReQU networks: if the measured expected $H^1\\times H(\\mathrm{div})$ error does not decay at least as $C(1/m+1/\\sqrt N)$ with $C$ polynomial in $d$, the claimed dimension-exempt rate fails. Concretely, take $u^*=\\cos(\\pi x_1)$ on $[0,1]^d$ with Neumann boundary data, train the first-order system with increasing $m$ and $N$, and check that the squared error obeys the predicted $1/m+1/\\sqrt N$ scaling; a slower decay for fixed $d$ contradicts the theorem.","tokens_in":31599,"feed_emoji":"📐","tokens_out":9254,"duration_ms":81045,"temperature":0.7,"pith_summary":"The paper establishes a priori error estimates for the deep mixed residual method (MIM) applied to $2n$-th order elliptic equations on the unit cube, with non-homogeneous Dirichlet, Neumann, or Robin boundary data. The central claim is that when the exact solution lies in a spectral Barron space, the expected squared error of the MIM approximations to the solution and its Laplacian derivatives is bounded by $O(\\|u^*\\|_{B}^2/m + \\|u^*\\|_{B}^2/\\sqrt N)$ for the Neumann and Robin cases, and by $O(\\|u^*\\|_{B}^2/\\sqrt m + \\|u^*\\|_{B}^2/N^{1/4})$ for the Dirichlet case; here $m$ is the network width and $N$ the number of interior collocation points. The constants are at most polynomial in the dimension, so the bound does not suffer the curse of dimensionality. The proof splits the error into approximation, generalization, and optimization components, using Céa's lemma, and controls the first two by Barron-space approximation and Rademacher complexity respectively. A side consequence is that MIM needs only quadratic or cubic ReLU activations, whereas the deep Ritz method needs ReLU of order $n+1$ for the same equation.","feed_headline":"MIM error bound escapes the curse of dimensionality","feed_subtitle":"Two-layer networks solve high-order elliptic PDEs at a rate that barely sees the dimension.","key_machinery":"The load-bearing object is the bilinear form $B_\\alpha(u,w)=(Pu,Pw)_{L^2(\\Omega)}+\\lambda(S_\\alpha u,S_\\alpha w)_{L^2(\\partial\\Omega)}$ (plus a zero-mean integral term for Neumann), where $P$ is a first-order differential operator coupling $\\nabla\\phi_k$ to $\\psi_k$ and $\\mathrm{div}\\,\\psi_k$ to $\\phi_{k+1}$, and $S_\\alpha$ is the boundary trace operator for Dirichlet, Neumann, or Robin data; the second-order system uses the analogous operator $P^*$ with $\\Delta\\phi_k-\\phi_{k+1}$. Coercivity of $B_\\alpha$ in the $H^1\\times H(\\mathrm{div})$ norm is proven by a perturbation technique: a sequence of small parameters $\\delta_k$ weights the cross terms so that Young's inequality absorbs them into the decoupled energy. The Dirichlet case, where coercivity in $H^1$ is impossible, uses a sup-linear coercivity estimate $\\sqrt{B_D(u,u)}\\,E \\gtrsim \\|u\\|^2_Z$, which after Céa's lemma turns the rates into square roots.","core_discovery":"The paper proves that the MIM empirical risk minimizer converges to the true solution tuple $(\\phi_k,\\psi_k)\\approx(\\Delta^k u^*,\\nabla\\Delta^k u^*)$ at a rate that is additive in the approximation and generalization errors. For the first-order least-squares system with ReQU networks, assuming $u^*\\in B^{2n+2}$, the expectation over Monte Carlo samples satisfies $\\sum_{k=0}^{n-1}(E\\|\\hat\\phi_k-\\Delta^k u^*\\|^2_{H^1} + E\\|\\hat\\psi_k-\\nabla\\Delta^k u^*\\|^2_{H(\\mathrm{div})}) \\lesssim \\|u^*\\|^2_{B^{2n+2}}/m + \\|u^*\\|^2_{B^{2n+2}}/\\sqrt N$ for Neumann and Robin conditions. For Dirichlet boundary data, the standard $L^2$ penalty causes a loss of $3/2$ in Sobolev regularity, so the paper derives a sup-linear coercivity for the bilinear form and obtains the square-root rate $\\sqrt{E_{\\mathrm{app}}}+\\sqrt{E_{\\mathrm{gen}}}$, giving $O(1/\\sqrt m + N^{-1/4})$. The second-order system with ReCU networks repeats the same structure under the stronger assumption $u^*\\in B^{2n+3}$, yielding analogous estimates for $\\hat\\phi_k-\\Delta^k u^*$ in $H^1$ and $\\nabla\\hat\\phi_k-\\nabla\\Delta^k u^*$ in $H(\\mathrm{div})$.","pith_inferences":["An implication left implicit in the paper is that the practical rate of MIM depends on the optimizer's ability to reach a global minimizer of the nonconvex empirical loss, so the theorem is best read as a guarantee on the landscape rather than on gradient descent.","Because the rate improvement over deep Ritz comes from the mixed formulation's lower-order loss, the same Barron/Céa/Rademacher decomposition should apply to other mixed least-squares neural methods, for instance first-order systems for elasticity or Stokes flow.","The Dirichlet square-root rate suggests that any penalty-free exact boundary enforcement in the spirit of the original MIM papers might restore the $O(1/m)$ rate, but the present analysis deliberately works with the $L^2$ penalty and does not make that claim."],"forward_implications":["For Neumann and Robin boundary data, the theorem predicts that the expected squared $H^1(\\Omega)\\times H(\\mathrm{div};\\Omega)$ error over all Laplacian iterates decays as $O(\\|u^*\\|^2_B/m + \\|u^*\\|^2_B/\\sqrt N)$, with an additional boundary sample count $N_\\partial = O(N^{d/2})$ keeping the boundary contribution from spoiling this rate.","For Dirichlet data the same error decays only as $O(\\|u^*\\|^2_B/\\sqrt m + \\|u^*\\|^2_B/N^{1/4})$, so the boundary penalty costs a square root in both network width and sample count.","MIM requires only ReQU (quadratic) activations for the first-order formulation and ReCU (cubic) activations for the second, while the deep Ritz method needs $\\mathrm{ReLU}^{n+1}$; for high-order equations this is a large reduction in required activation smoothness.","The bounds are statements about expectation with respect to the random training samples and about global minimizers of the empirical loss; optimization error is not analyzed, so a training procedure that fails to reach the global minimum can produce larger errors than the theorem bounds."],"supporting_citations":[{"why":"Supplies the ReQU approximation theorem in Barron space, the sup-linear coercivity technique for the Dirichlet boundary, and the Rademacher-complexity lemmas on which the estimates are built.","marker":"[13]"},{"why":"Provides the abstract bilinear-form/Céa framework that the paper uses to decompose total error into approximation, generalization, and optimization errors.","marker":"[32]"},{"why":"Introduces the first-order system least-squares formulation that MIM's first-order loss (2) is based on.","marker":"[4]"},{"why":"Defines the spectral Barron space and gives the a priori generalization analysis for deep Ritz that the approximation argument extends.","marker":"[16]"},{"why":"Barron's convex-hull approximation theorem is the engine behind the $O(B^2/m)$ network-width rate used in Appendix A.","marker":"[2]"},{"why":"Establishes Barron-space approximation for ReLU$^k$ networks; the paper sharpens the constant to be dimension-independent.","marker":"[14]"},{"why":"Documents the $3/2$ regularity loss for $L^2$ Dirichlet penalties, which motivates the sup-linear coercivity route.","marker":"[26]"}],"fun_headline_variants":["MIM error bound escapes curse of dimensionality","Provable MIM error rates for high-order elliptic PDEs","MIM tames high-order elliptic error without bottleneck","No curse of dimensionality in MIM error analysis","MIM reduces regularity demands for high-order PDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact solution is assumed to belong to a spectral Barron space $B^{2n+2}$ or $B^{2n+3}$, and the proof uses that assumption both for the $1/m$ approximation rate and for the a priori parameter bounds in the network class; if a solution of the elliptic equation is not Barron-regular, the dimension-exempt conclusion has no basis here.","fun_headline_variants_meta":{"raw":{"variants":["MIM error bound escapes curse of dimensionality","Provable MIM error rates for high-order elliptic PDEs","MIM tames high-order elliptic error without bottleneck","No curse of dimensionality in MIM error analysis","MIM reduces regularity demands for high-order PDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1579,"prompt_tokens":1007,"completion_tokens":572,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":507}},"tokens_in":623,"tokens_out":572,"duration_ms":5667,"temperature":1.0,"reasoning_tokens":507,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:29:54.313084+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a sequence of exact solutions of the elliptic equation whose Barron norm grows with the dimension but whose classical Sobolev regularity is fixed, and run MIM with ReQU networks: if the measured expected $H^1\\times H(\\mathrm{div})$ error does not decay at least as $C(1/m+1/\\sqrt N)$ with $C$ polynomial in $d$, the claimed dimension-exempt rate fails. Concretely, take $u^*=\\cos(\\pi x_1)$ on $[0,1]^d$ with Neumann boundary data, train the first-order system with increasing $m$ and $N$, and check that the squared error obeys the predicted $1/m+1/\\sqrt N$ scaling; a slower decay for fixed $d$ contradicts the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ReQU approximation theorem in Barron space, the sup-linear coercivity technique for the Dirichlet boundary, and the Rademacher-complexity lemmas on which the estimates are built."},{"cited_title":"A Unified Framework for the Error Analysis of Physics-Informed Neural Networks","cited_arxiv_id":"2311.00529","evidence_quote":"Provides the abstract bilinear-form/Céa framework that the paper uses to decompose total error into approximation, generalization, and optimization errors."},{"cited_title":"6, 1785–1799","cited_arxiv_id":null,"evidence_quote":"Introduces the first-order system least-squares formulation that MIM's first-order loss (2) is based on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the spectral Barron space and gives the a priori generalization analysis for deep Ritz that the approximation argument extends."},{"cited_title":"3, 930–945","cited_arxiv_id":null,"evidence_quote":"Barron's convex-hull approximation theorem is the engine behind the $O(B^2/m)$ network-width rate used in Appendix A."},{"cited_title":"1, 319–344","cited_arxiv_id":null,"evidence_quote":"Establishes Barron-space approximation for ReLU$^k$ networks; the paper sharpens the constant to be dimension-independent."},{"cited_title":"1, 47–69","cited_arxiv_id":null,"evidence_quote":"Documents the $3/2$ regularity loss for $L^2$ Dirichlet penalties, which motivates the sup-linear coercivity route."}],"review_version":1}