An Iterative Deep Ritz Method for Monotone Elliptic Problems

Bangti Jin; Fengru Wang; Tianhao Hu

REVIEW 4 major objections 6 minor 54 references

An Iterative Deep Ritz Method for Monotone Elliptic Problems

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.15186 v1 pith:W5OVWD2J submitted 2025-01-25 math.NA cs.LGcs.NA

Tianhao Hu , Bangti Jin , Fengru Wang This is my paper · ORCID

classification math.NAcs.LGcs.NA MSC 65N1265N3068T0747H05

keywords iterativedeepRitzmethodmonotoneellipticoperatorsneuralPDEsolversconvergenceanalysisBanachspacegeometrydualitymaplearningerrorp-Laplaceequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

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.

Referee Report

4 major / 6 minor

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.

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 (4)

[Section 4, Assumption 4.1] 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 6.1, Eq. (6.13)] 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.
[Theorem 4.1 vs. Section 5] 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.
[Theorems 4.1 and 4.2, ϵ_M] 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.

minor comments (6)

[Section 3.2] 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').
[References] 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.
[Example 5.2(ii)] 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.
[Tables 2 and 3] 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.
[Algorithm 1, line 8] 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 6.1, Step 4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence analysis is a conditional a priori Lyapunov argument, and the main gaps are unproved boundedness and an analyzed-versus-implemented step-size mismatch rather than circular reductions.

full rationale

The paper's central claim is Theorem 4.1, which states that under Assumptions 2.1 and 4.1 the IDRM iterates converge to the unique solution u* with a preasymptotic rate. The proof is a self-contained Lyapunov argument: Lemma 6.1 derives local uniform monotonicity and continuity of Dmu from Assumption 2.1; Proposition 6.3 gives a unique local minimizer; Lemma 6.2 proves two-sided bounds between ||u-u*||_X and the dual potential Phi*_mu(A(u)-f); and the recurrence (6.17) is obtained algebraically from these bounds, the inexact update inequality, the learning-error tolerance, and the chosen lambda_k schedule. No constant in the theorem is fitted to numerical data, and no benchmark error is used as an input to the derivation. The step-size identity (4.1), while superficially self-referential because lambda_k appears in L_k and also in the proposed formula through v_{k+1}, is resolved inside the paper: -L_k(v_{k+1}) = Phi*_mu(f-A(u_k)), which is independent of lambda_k. Thus this is not a definitional circle.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the abstract elliptic-operator assumptions (Assumption 2.1) and on an unproven boundedness of the iterates (Assumption 4.1). The learning error bound assumes a global empirical minimizer, which the training heuristics do not guarantee. No new physical entities are introduced.

free parameters (4)

convexity penalty μ = 0, 0.01, 2 (per example)
Chosen by hand for each experiment; controls the local convexity neighborhood Vμ in (3.4) and the convergence constants; no automatic selection rule is given.
initial step size λ0 = 1 in all experiments
Set to 1; the theoretical schedule λk = cs Φ*_μ(f - A(uk))^α from Theorem 4.1 is not described in the experiments.
finite-difference smoothing δ (Example 5.2(ii)) = 0.01
Used to approximate the singular duality product for p=1.5; the effect on accuracy is not analyzed.
boundary penalty continuation parameters (Example 5.4) = σ1=10, ρ=1.5
Hand-tuned path-following schedule for Navier-Stokes; not covered by the convergence theory.

assumptions (5)

domain assumption Assumption 2.1: A is continuous, Gateaux differentiable, satisfies (2.2), coercive, locally uniformly monotone, and the asymmetric part is Lipschitz
Defines the class of elliptic operators. Verified for linear elliptic, p-Laplace, and Navier-Stokes in Examples 2.1-2.3, but not for the parabolic example 5.3.
ad hoc to paper Assumption 4.1: the iterates {uk} are uniformly bounded in X
This boundedness is assumed without proof and is used throughout the convergence analysis (e.g., Lemma 6.1 and recurrence (6.17)). It is a load-bearing premise that is not established for the algorithm.
domain assumption Assumption 4.2: Nemytskii growth on a_m and L∞ boundedness of sources
Used in the statistical error bound (Proposition 6.5) to control the loss integrands; reasonable for the examples.
domain assumption The DNN class N is contained in X = W^{1,p}(Ω) and the loss is differentiable in the DNN parameters
Requires tanh activation (excludes ReLU) and ensures the empirical minimization is well-defined, as stated in Section 3.2.
ad hoc to paper The empirical loss bLk is globally minimized over N (or the tolerance ϵM accounts for the optimization gap)
Theorem 4.2 bounds the learning error for the empirical minimizer, but the training uses ADAM which is not guaranteed to reach a global minimum; the optimization error is not analyzed.

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Pith. "Pith review of An Iterative Deep Ritz Method for Monotone Elliptic Problems." pith.science (2026). https://pith.science/paper/W5OVWD2J

@misc{pith2026250115186,
  author       = {Pith},
  title        = {Pith review of: An Iterative Deep Ritz Method for Monotone Elliptic Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5OVWD2J}},
  note         = {Machine review of arXiv:2501.15186}
}

read the original abstract

In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.

Figures

Figures reproduced from arXiv: 2501.15186 by the authors.

**Figure 1.** The DNN approximations for Example 5.1, slices at xi = 1 2 (i = 3, 4, · · · , 10). (top: IDRM, middle: WAN, bottom: PINN) To shed further insights, we show in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

**Figure 2.** The training dynamics of IDRM for Example [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** The DNN approximation for Example 5.2(i), slices at xi = 1 2 (i = 3, 4, · · · , 10). 0 500 1000 1500 2000 i 0 1 2 3 4 5 6 loss 0 500 1000 1500 2000 i 0.1 0.2 0.3 0.4 0.5 0.6 0.7 error (a) L k σ vs i (b) e vs i [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The training dynamics of IDRM for Example [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: The DNN approximations for Example 5.2(ii), slices at xi = 1 2 (i = 3, 4, · · · , 10) (top: IDRM, middle: WAN, bottom: PINN). 0 500 1000 1500 2000 2500 3000 i 0 2 4 6 8 10 12 loss 0 500 1000 1500 2000 2500 3000 i 0.1 0.2 0.3 0.4 0.5 0.6 error (a) L k σ vs i (b) e vs i …

**Figure 6.** Figure 6: The training dynamics of IDRM for Example [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: The DNN approximation for Example 5.3, slices at xi = 1 2 (i = 3, 4, · · · , 10), at t = 0.3. 0 0.1 0.2 0.3 0.4 0.5 t 0 1 2 3 4 5 6 7 8 error 10-3 0 0.1 0.2 0.3 0.4 0.5 t 2.46 2.48 2.5 2.52 2.54 2.56 2.58 2.6 Function value exact predict (a) e vs t (b) e0 vs t [PITH_F…

**Figure 8.** Figure 8: (a) The evolution of the relative error eidrm, (b) the variation of the exact and predicted function values at x = (1/2, 1/2, 1/2, · · · , 1/2). The last example is about the three-dimensional Navier-Stokes problem. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: with a relative error eidrm = (2.85e-2, 2.76e-2, 1.90e-2)⊤ for the three components, indicating the good accuracy of the DNN approximation. -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.005 0.01 0.015 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -0.6 -0…

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