{"id":"b37988bf-85d4-42c7-90a2-51a25363196f","arxiv_id":"2501.15186","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"An iterative deep Ritz method for monotone elliptic problems, with convergence and learning-error bounds, outperforms PINN and WAN on nonsmooth benchmarks.","lead":"This paper presents an iterative deep Ritz method that solves elliptic equations with monotone, possibly non-symmetric operators by repeatedly minimizing a local convex surrogate loss. It proves preasymptotic convergence rates and shows better accuracy than PINN and weak adversarial networks on several nonsmooth, high-dimensional benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1 rests on Assumption 4.1 (uniform boundedness of iterates), which is neither proved nor numerically verified; the implemented λ/constraint scheme also differs from the analyzed one.","rationale":"The reader's weakest_assumption and my concern coincide: Assumption 4.1 is the load-bearing point. I agree with the conditional verdict. The theorem states the assumption explicitly, so the mathematical statement is not false; the issue is that the paper uses the theorem as a convergence guarantee for the practical algorithm without supplying a proof or an experimental check of boundedness. My stress-test also notes that the inexact condition (3.6) does not constrain u_{k+1} to u_k+Vμ, so the Bregman-distance argument in Step 1 appears to require a stronger condition than Algorithm 1 actually enforces; this strengthens rather than replaces the reader's concern. The step-size and dual-potential issue is real but secondary: Remark 4.1 shows λ_k is independent of v_{k+1} in principle, but the paper gives no way to compute Φ*_μ efficiently, and the implementations in Section 5 do not use the schedule. I therefore would not change the CONDITIONAL verdict: the central idea is promising and the analysis is extensive, but the main convergence guarantee is conditional on an unverified uniform bound, and the numerical results do not close that gap. A norm-logging study or a targeted exact-update simulation would settle whether the assumption is actually satisfied for the model problems considered.","tokens_in":33105,"tokens_out":13970,"duration_ms":127454,"concrete_test":"With the released code, log ∥u_k∥_{W^{1,p}}, ∥u_{k+1}-u_k∥, and the actual λ_k for Examples 5.1–5.4. If max_k ∥u_k∥_{W^{1,p}} exceeds the radius R required for the constants in Lemmas 6.1–6.2, or if the norm grows without bound while (6.17) predicts decay, Assumption 4.1 fails in the reported regime. Separately, run a 1D finite-element version of the exact update u_{k+1}=u_k+λ_k^{-1}D_μ^{-1}(f-A(u_k)) for p-Laplace with convection under the Theorem 4.1 λ-rule; an admissible unbounded run shows Assumption 4.1 is a genuinely extra hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence claim (Theorem 4.1) is conditional on Assumption 4.1, the uniform boundedness of {u_k} in X=W^{1,p}. This is not a harmless technicality. Algorithm 1's inexact condition (3.6) only requires L_k(u_{k+1}) ≤ min_{u∈u_k+Vμ} L_k(u)+ϵ_M; it does not require u_{k+1}∈u_k+Vμ. Step 1 of the proof then applies the Bregman-distance lower bound (6.8) to λ_k(u_{k+1}-u_k) and λ_k(v_{k+1}-u_k), which is only known to hold when both arguments lie in Vμ; the iterates produced by (3.6) need not satisfy this. Even if one reads line 5 as a constrained minimization, the constraint ball is centered at the current iterate, so ∥u_k∥≤∥u_0∥+k·rad(Vμ) grows linearly; uniform boundedness does not follow from the displayed algorithm. No argument in Section 6 establishes Assumption 4.1, and the numerical section never reports ∥u_k∥_{W^{1,p}}, so the assumption is unchecked for the examples that support the paper's claims. All constants entering the recurrence (6.17) — c_p, c_r, c_s, the local monotonicity constant in Lemma 6.1 and the norm equivalences in Lemma 6.2 — depend on the common radius R; if no such R exists, the monotone decay in (i) and the exponential/algebraic rates in (ii) cannot be concluded. A secondary gap: Theorem 4.1 chooses λ_k=c_sΦ*_μ(f-A(u_k))^α, but Algorithm 1 gives no implementable rule for evaluating the dual potential, and Section 5 fixes λ at 1 or uses heuristic updates, so the analyzed iteration is not the one benchmarked. The theorem itself is honestly conditional, but the paper's claim that IDRM provides a convergence rate gains support only if Assumption 4.1 is either proved or verified in experiments.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces IDRM, an iterative neural-network solver for monotone elliptic problems A(u)=f on W^{1,p}(Ω), 1<p≤2 (and via Corollary 6.1 for p>2 in a special case). Each outer iteration k defines a local surrogate loss L_k(u)=Φ_μ[λ_k(u-u_k)]+⟨λ_k(u-u_k),A(u_k)-f⟩, where Φ_μ is a convexified potential with penalty μ, and minimizes it (inexactly) over a DNN class. The main result (Theorem 4.1) states that, under Assumptions 2.1 and 4.1 (uniform boundedness of iterates), the dual potential Φ*_μ(f-A(u_k)) decreases monotonically and the error decays exponentially (p=ρ=2) or algebraically, with rates depending on the learning error ϵ_M. Theorem 4.2 bounds the expected learning error in terms of DNN width, depth, and sample size. Numerical experiments on a 10D convection-diffusion problem, p-Laplace problems (p=2.5 and p=1.5), a parabolic problem, and 3D Navier-Stokes show IDRM outperforming PINN and WAN.","tokens_in":33587,"tokens_out":12368,"duration_ms":101483,"significance":"If the hypotheses can be established, IDRM is a significant addition to neural PDE solvers: it handles non-symmetric monotone operators without a global Ritz potential, requires only weak solutions, and avoids the saddle-point instability of WAN. The theoretical framework relying on Bregman distances and dual potentials is original, and the error analysis using Rademacher complexity is nontrivial. The numerical results are promising, but the current gap between the analyzed and implemented algorithms, and the unverified boundedness assumption, prevent the paper from fully supporting its advertised convergence guarantee.","major_comments":[{"comment":"Assumption 4.1 is not proved and not verified numerically. It is load-bearing: Lemma 6.1's constants, Lemma 6.2's norm equivalences, and the recurrence (6.17) all require a common radius R bounding every iterate. Algorithm 1 does not enforce such a bound; even if line 5 is read as constrained minimization over u_k+V_μ, the radius of V_μ is fixed, so ∥u_k∥_X can grow linearly in k. Section 5 reports only relative L^2 errors and never reports ∥u_k∥_{W^{1,p}}, so the assumption is unchecked for all examples. Please either prove the bound for a modified algorithm or check it empirically and state Theorem 4.1 as conditional.","section":"Section 4, Assumption 4.1"},{"comment":"The proof of the bound ∥u_{k+1}-v_{k+1}∥_X ≤ λ_k^{-1}ρ^{1/ρ}ϵ_M^{1/ρ} requires the Bregman-distance inequality (6.8) for the pair (λ_k(u_{k+1}-u_k), λ_k(v_{k+1}-u_k)). However, (6.8) is proved only for arguments in V_μ (Proposition 6.3). The second argument is in V_μ by definition of v_{k+1}, but the first need not be, because condition (3.6) in Algorithm 1 does not require u_{k+1}∈u_k+V_μ; it only bounds L_k(u_{k+1}) by the minimum over u_k+V_μ plus ϵ_M. Please add the constraint u_{k+1}∈u_k+V_μ to Algorithm 1 or supply an argument that the inexact minimizer automatically lies in the ball.","section":"Section 6.1, Eq. (6.13)"},{"comment":"The step-size schedule analyzed in Theorem 4.1, λ_k=c_sΦ*_μ(f-A(u_k))^α, is not used in the numerical experiments. Remark 4.1 asserts computability but gives no implementable procedure for evaluating the dual potential (a maximization over V_μ). Section 5 fixes λ_0=1 and, in Examples 5.1, 5.2(i), 5.3, and 5.4, sets μ=0; no λ_k update is performed. Thus the numerical results in Section 5 concern a different iteration than the one for which convergence is proved. The authors should either implement and report the theoretical schedule (or a justified approximation) or analyze the heuristic schedule actually used.","section":"Theorem 4.1 vs. Section 5"},{"comment":"Theorem 4.2 bounds the expected learning error EX,T[ϵ_M], while Theorem 4.1 treats ϵ_M as a deterministic constant in the recurrence (6.17). To combine the two results—for instance, to conclude that IDRM attains a prescribed accuracy with high probability—a high-probability version of Theorem 4.2 (or a concentration argument) is needed. The manuscript does not provide this link. Please clarify whether Theorem 4.2 is intended as input to Theorem 4.1 and, if so, add the missing probabilistic transfer.","section":"Theorems 4.1 and 4.2, ϵ_M"}],"minor_comments":[{"comment":"The phrase 'we start the training afresh, using the DNN parameter from the last loop to warm-start the current optimization process' is contradictory; 'afresh' should be replaced (e.g., 'we start each training from the previous loop's parameters').","section":"Section 3.2"},{"comment":"Reference [45] has garbled author names ('C. ronstein, Michael M. ad Lemieux'); please correct the citation to the actual authors of the message-passing Monte Carlo paper.","section":"References"},{"comment":"The finite-difference smoothing parameter δ=0.01 is introduced to handle the singularity of the p-Laplace duality product, but no sensitivity study or convergence test with respect to δ is reported; the accuracy claim depends on this ad-hoc parameter.","section":"Example 5.2(ii)"},{"comment":"The comparisons report single runs without standard deviations or multiple random seeds; given the stochastic nature of ADAM training, the claimed accuracy improvements over PINN and WAN should be supported by repeated trials.","section":"Tables 2 and 3"},{"comment":"The stopping criterion 'while −L_k(u_{k+1}) ≤ ϵ_tol' is unclear; since L_k can be negative and changes with k, please specify how this relates to the thresholds k* and k** in Theorem 4.1.","section":"Algorithm 1, line 8"},{"comment":"Several displayed equations in the proof have broken exponents and missing superscripts (e.g., the line containing 'Φ∗ µ(pk)− αp(p−1) ρ−1 ϵ...'), making the argument hard to follow; please typeset the exponents correctly.","section":"Section 6.1, Step 4"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of math.NA and the methodology is of interest. The main concern is that Theorem 4.1 is conditional on an unverified assumption and the numerical algorithm differs from the analyzed one; these are fixable but require substantial revision. I recommend major revision rather than rejection because the core idea and the proof machinery are promising."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the algorithmic idea is genuinely new: they rewrite a monotone, possibly non-symmetric elliptic problem A(u)=f as a sequence of local convex minimizations, using the current residual A(u_k)-f as the driving term and adding a convex penalty to keep each update in a region where a local potential exists. This extends deep Ritz to problems without a global Ritz potential, which covers convection-diffusion, p-Laplace, and Navier-Stokes. That is a real advance over PINN, DRM, and WAN. Second, the headline convergence result, Theorem 4.1, is explicitly conditional on Assumption 4.1: the iterates {u_k} must stay uniformly bounded in W^{1,p}. They never prove that assumption and never numerically verify it. Worse, the algorithm as written in line 5 of Algorithm 1 only requires L_k(u_{k+1}) to beat the minimum over u_k+V_mu within tolerance epsilon_M; it does not require u_{k+1} itself to lie in u_k+V_mu. The proof's Bregman-distance lower bound is only valid when the arguments are in V_mu, so the produced iterates need not satisfy the key inequality. Even reading line 5 as a constrained minimization, the constraint ball is centered at the current iterate, so its radius grows linearly with k and gives no uniform boundedness. On top of this, the step-size rule in Theorem 4.1, lambda_k = c_s Phi*_mu(f-A(u_k))^alpha, is never implemented; the experiments fix lambda_0=1 and use constant or heuristic updates, and several examples set mu=0, which is only covered by a separate corollary for operators of the form DPhi + linear skew. So the analyzed iteration is not the one benchmarked. The numerical results are encouraging but are single runs without error bars; they illustrate the method rather than confirm the rates. What is done well: the analysis is serious and detailed. They bring in duality maps, Bregman distances, local monotonicity, and derive preasymptotic rates in both Hilbert and Banach settings. The learning-error analysis with DNN approximation and Rademacher complexity is thorough and mostly standard. The experiments cover genuinely challenging cases, including a 10D convection-diffusion problem with a nonsmooth solution and a 3D Navier-Stokes problem, and IDRM reports better accuracy than PINN and WAN on those examples. The paper is worth a serious referee. The gaps are real but tractable: Assumption 4.1 could be proved for the actual algorithm or replaced by a reasonable condition, the algorithm/proof mismatch can be fixed by writing the constraint in the scaled ball, and the experiments can add norm checks and the theoretical step-size schedule. I would send it to peer review, expecting revision.","headline":"A genuinely new iterative deep Ritz method for monotone non-symmetric elliptic problems, with a serious convergence analysis that is conditional on an unproven boundedness assumption and a step-size rule that the experiments do not implement.","tokens_in":34085,"tokens_out":4120,"would_cite":true,"duration_ms":38767,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N12","65N30","68T07","47H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"IDRM, an iterative deep Ritz method, converges to the unique weak solution of monotone elliptic problems—non-symmetric ones included—with exponential or algebraic preasymptotic rates and a learning error controllable by network size and…","keywords":["iterative deep Ritz method","monotone elliptic operators","neural PDE solvers","convergence analysis","Banach space geometry","duality map","learning error","p-Laplace equation"],"falsifier":"Compute the dual-potential sequence $\\Phi_\\mu^*(f-A(u_k))$ during an IDRM run on a monotone problem with a known solution. Theorem 4.1 predicts this sequence decreases monotonically until it reaches the threshold $\\Delta_c$; a single step with bounded iterates where $\\Phi_\\mu^*(p_{k+1})>\\Phi_\\mu^*(p_k)$ while $-L_k(v_{k+1})>\\Delta_c$ would falsify the central convergence claim.","tokens_in":32912,"feed_emoji":"🧮","tokens_out":15536,"duration_ms":119897,"temperature":0.7,"pith_summary":"The paper introduces a neural-network solver, the iterative deep Ritz method (IDRM), for elliptic equations $A(u)=f$ whose operator is monotone but need not come from a variational principle. Rather than minimizing one global Ritz functional, IDRM solves a sequence of local convex surrogate problems, each built around the current approximation, so non-symmetric and low-regularity problems are within reach. The paper aims to prove that—provided the iterates stay bounded—the method converges to the unique weak solution, with exponential error decay when the underlying space is Hilbert ($p=\\rho=2$) and algebraic decay in the Banach case, down to a level set by the learning error. It also proves the learning error can be driven to zero by wider networks and more sampling points. This matters because standard deep Ritz needs a symmetric operator and PINN needs strong solutions, while IDRM claims neither.","feed_headline":"Provable convergence for monotone elliptic PDEs via iterative deep Ritz","feed_subtitle":"A neural solver that needs no Ritz potential or strong solutions, with exponential or algebraic error rates.","key_machinery":"The load-bearing construction is the local surrogate loss $L_k(u)=\\Phi_\\mu[\\lambda_k(u-u_k)]+\\langle\\lambda_k(u-u_k),A(u_k)-f\\rangle$, where $\\Phi_\\mu(u)=\\int_0^1\\langle u,A(tu)\\rangle\\,dt+\\mu\\|u\\|_X^2$. This functional is locally convex inside the ball $V_\\mu=\\{u:\\|u\\|_X\\le 3\\mu(p-1)/(2N)\\}$, so each inner step has a unique minimizer $v_{k+1}$, and the dual potential $\\Phi_\\mu^*$ makes $D\\Phi_\\mu^*$ act as an approximate inverse of $A$. The proof chains the Bregman-distance lower bound from local strong convexity, norm equivalence via $\\Phi_\\mu^*$, and the recursion $\\Phi_\\mu^*(p_{k+1})\\le c_r\\varepsilon+\\Phi_\\mu^*(p_k)-c_p\\Phi_\\mu^*(p_k)^\\beta$, from which the exponential and algebraic rates follow.","core_discovery":"The central claim is Theorem 4.1: under Assumptions 2.1 and 4.1, the IDRM iterates $u_k$ generated by Algorithm 1, with step size $\\lambda_k = c_s \\Phi_\\mu^*(f-A(u_k))^\\alpha$, converge to the unique solution $u^*$ of the monotone elliptic problem (2.1). In the Hilbert case $p=\\rho=2$ the error decays as $\\|u_k-u^*\\|_X \\le (\\eta q)^k \\|u_0-u^*\\|_X$; in the Banach case with $0<p(p-1)-(\\rho-1)^2<1$ it decays algebraically as $\\|u_k-u^*\\|_X \\le C((1-\\eta)k)^{-1/(\\beta-1)}$. Theorem 4.2 shows the per-iteration learning error can be made arbitrarily small by choosing a sufficiently wide tanh network and sufficiently many sampling points. Together these results assert that a neural PDE solver for monotone, possibly non-symmetric elliptic operators can converge without a global Ritz potential and without high solution regularity.","pith_inferences":["The paper states the step-size rule $\\lambda_k=c_s\\Phi_\\mu^*(f-A(u_k))^\\alpha$ in the theorem, but the numerical experiments fix $\\lambda_0=1$ and often take $\\mu=0$; testing the proven rule against the hand-tuned version would show how much of the observed accuracy is due to the scheduled step size.","The learning-error analysis leaves the inner-loop optimization error inside $\\epsilon_M$; separating optimization error from approximation and sampling error is a natural next step.","Assumption 4.1 might be provable from the surrogate construction itself: a contraction-type estimate on each step would turn the assumed uniform boundedness into a theorem.","The method's preconditioning interpretation—that $D\\Phi_\\mu^*$ approximates $A^{-1}$—suggests IDRM could be compared quantitatively with classical fixed-point and preconditioned residual iterations on the same monotone problems."],"forward_implications":["IDRM extends deep Ritz methods to non-symmetric monotone problems such as convection-dominated advection-diffusion, p-Laplace, and steady Navier-Stokes equations.","Because only first weak derivatives enter the surrogate loss, the method can approximate solutions in $W^{1,p}(\\Omega)$ that strong-form solvers such as PINN cannot handle at the population level.","The convergence is preasymptotic: exponential in the Hilbert case and algebraic in the Banach case, with a precision floor set by the learning error.","The learning error can be made arbitrarily small by increasing DNN width (with logarithmic depth) and the number of sampling points, so total error is controlled by network and sampling capacity.","Each inner problem is convex in the function $u$, which makes the local minimization better conditioned than the non-convex global Ritz or adversarial losses."],"supporting_citations":[{"why":"It supplies existence and uniqueness of the inverse operator $A^{-1}$, its uniform continuity, and the topological fixed-point argument proving each surrogate loss has a unique minimizer.","marker":"[44]"},{"why":"It provides the duality-map inequality used to establish local monotonicity of $D_\\mu$ in Lemma 6.1.","marker":"[52]"},{"why":"It defines the deep Ritz variational minimization that IDRM iterates and extends to non-symmetric problems.","marker":"[17]"},{"why":"It is the strong-form PINN baseline whose high-regularity requirement IDRM avoids and against which IDRM is compared numerically.","marker":"[41]"},{"why":"It describes the weak adversarial formulation that handles non-symmetric problems and is IDRM's main numerical competitor.","marker":"[53]"},{"why":"It gives the tanh-network Sobolev approximation bound used in Theorem 4.2 for the approximation component of the learning error.","marker":"[20]"},{"why":"It provides Rademacher-complexity estimates for DNN function classes used to bound the statistical component of the learning error.","marker":"[26]"},{"why":"It supplies the $W^{1,\\infty}$ norm bound and derivative-network Rademacher estimates used in Proposition 6.5.","marker":"[27]"}],"fun_headline_variants":["Neural solver provably converges for monotone elliptic PDEs","Iterative deep Ritz method guarantees convergence without Ritz potential","Provable error decay for monotone PDEs via iterative deep Ritz","Monotone elliptic problems solved without strong solutions, with rates","Deep Ritz iterative method handles non-symmetric operators with proofs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the iterates $u_k$ always stay inside one fixed ball in the Sobolev space $W^{1,p}(\\Omega)$, a boundedness condition stated as Assumption 4.1 without proof or numerical check and on which every local convexity constant and decay estimate depends.","fun_headline_variants_meta":{"raw":{"variants":["Neural solver provably converges for monotone elliptic PDEs","Iterative deep Ritz method guarantees convergence without Ritz potential","Provable error decay for monotone PDEs via iterative deep Ritz","Monotone elliptic problems solved without strong solutions, with rates","Deep Ritz iterative method handles non-symmetric operators with proofs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1357,"prompt_tokens":943,"completion_tokens":414,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":326}},"tokens_in":559,"tokens_out":414,"duration_ms":3576,"temperature":1.0,"reasoning_tokens":326,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:32:03.581466+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the dual-potential sequence $\\Phi_\\mu^*(f-A(u_k))$ during an IDRM run on a monotone problem with a known solution. Theorem 4.1 predicts this sequence decreases monotonically until it reaches the threshold $\\Delta_c$; a single step with bounded iterates where $\\Phi_\\mu^*(p_{k+1})>\\Phi_\\mu^*(p_k)$ while $-L_k(v_{k+1})>\\Delta_c$ would falsify the central convergence claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It describes the weak adversarial formulation that handles non-symmetric problems and is IDRM's main numerical competitor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies existence and uniqueness of the inverse operator $A^{-1}$, its uniform continuity, and the topological fixed-point argument proving each surrogate loss has a unique minimizer."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the duality-map inequality used to establish local monotonicity of $D_\\mu$ in Lemma 6.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It defines the deep Ritz variational minimization that IDRM iterates and extends to non-symmetric problems."},{"cited_title":"Guhring and M","cited_arxiv_id":null,"evidence_quote":"It gives the tanh-network Sobolev approximation bound used in Theorem 4.2 for the approximation component of the learning error."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides Rademacher-complexity estimates for DNN function classes used to bound the statistical component of the learning error."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the $W^{1,\\infty}$ norm bound and derivative-network Rademacher estimates used in Proposition 6.5."}],"review_version":1}