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REVIEW 4 major objections 4 minor 38 references

A deep first-order system least squares method for the obstacle problem

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves that a deep FOSLS neural-network method converges for obstacle problems whenever the sampled loss functions can be asymptotically minimized.

desk verdict A promising Deep FOSLS adaptation to obstacle problems, but the convergence theory only covers a(t)=ReLU while all experiments use t^2, so the central claim is not proven for the implemented method. read the letter →

arxiv 2508.19412 v1 pith:6CP6QKYW submitted 2025-08-26 math.NA cs.NA

Gabriel Acosta , Eugenia Belén , Francisco M. Bersetche , Juan Pablo Borthagaray This is my paper · ORCID
classification math.NAcs.NA MSC 49J4065N3068T07
keywords FOSLSobstacleproblemneuralnetworksGamma-convergencevariationalinequalitiesmesh-freemethodhigh-dimensionalLagrangemultiplier
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 presents a mesh-free neural-network method for the obstacle problem, which models the equilibrium of an elastic membrane constrained to stay above an obstacle. It reformulates the problem as a first-order least-squares minimization over triples consisting of the solution, its gradient, and a Lagrange multiplier, then parameterizes admissible triples by neural networks so that boundary and obstacle constraints are enforced exactly. The main theoretical result is that, under density and continuity assumptions on the network class, asymptotically minimizing the Monte Carlo sampled regularized functionals produces neural functions converging to the exact solution in the natural W-norm. This matters because it offers a convergence guarantee for a flexible, scalable, high-dimensional approach, with experiments reported in dimensions up to 20 and on a non-Lipschitz slit domain.

What carries the argument

The load-bearing object is the first-order least-squares functional L(w;f,g) = ||gamma+f||^2 + ||nabla u - phi||^2 + (gamma, u-g) + (phi, nabla(u-g)) on the admissible set KW = {(u,phi,gamma): u>=g, gamma>=div phi}, whose unique minimizer is p0 = (u0, nabla u0, -f). Around this, the method builds a parameterized admissible class Wm by writing u = g + d_∂Ω a(v), phi = psi, and gamma = div psi + a(eta), where d_∂Ω is a surrogate distance to the boundary and a is a Lipschitz nonnegative activation, so the constraints hold strongly for every parameter. Gamma-convergence of the sampled functionals, together with equi-coercivity, transfers asymptotic minimizers of the sampled loss to near-minimize

What would settle it

Take a problem with a known exact solution, such as Example 5.1, train with the architecture the proofs actually cover—one hidden layer, ReLU activation, a(t)=ReLU—and measure ||p_N - p_0||_W as width and sample size grow; if the error does not approach zero, the asymptotic-minimization premise of Theorem 4.3 fails for the trained parameters. A simpler probe: a(t)=t^2 is not Lipschitz on unbounded network outputs, so the boundedness lemma for the integrands can fail; checking whether computed loss values diverge with parameter magnitude would reveal the gap.

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Extended reading notes

Core claim

The central claim is Theorem 4.3: for every epsilon greater than zero, there exist a network width m, a parameter radius R, and a sample size N such that, P-almost surely, any sequence of parameters that asymptotically minimizes the sampled regularized functionals L_{xi,N} yields a neural function p_N whose W-distance to the exact obstacle solution p_0 is at most epsilon. The proof establishes coercivity and local Lipschitz continuity of the continuous least-squares functional L, shows that the Monte Carlo functionals Gamma-converge to L almost surely, and combines this with the fundamental theorem of Gamma-convergence to transfer near-minimizers of the sampled functionals to near-minimizers

Load-bearing premise

The theorem rests on the assumptions that the chosen neural-network class can approximate the exact solution in the W-norm arbitrarily well and that small parameter changes move the network function continuously; these are proved only for one-hidden-layer ReLU networks and a hybrid piecewise-constant network for the multiplier, not for the multi-layer SoftPlus networks and a(t)=t^2 used in the experiments.

Editorial extensions

If this is right

  • If the convergence theorem's hypotheses hold, the method is a provably convergent mesh-free solver: network width, parameter radius, and collocation sample size can be chosen to meet any prescribed accuracy in the W-norm, almost surely.
  • The simultaneous approximation of u, its gradient, and the Lagrange multiplier means a single training run delivers the solution, the flux, and an indicator of the contact set.
  • Because boundary and obstacle constraints are enforced in the network architecture rather than penalized, every candidate during optimization is admissible, which simplifies both the loss landscape and the interpretation of the minimizer.
  • The framework extends the Deep FOSLS method from linear elliptic PDEs to a variational inequality, showing that the Gamma-convergence argument survives the addition of a convex constraint.

Reading between the lines

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

  • A natural testable extension is to close the gap between theory and experiments: prove the density and continuity hypotheses for multi-layer SoftPlus networks and for activations such as a(t)=t^2 on bounded network outputs, then rerun the d=20 benchmark with the verified class.
  • The same admissible-parameterization plus Gamma-convergence template should apply to other variational inequalities, such as two-obstacle problems, Signorini contact, or parabolic obstacle problems, provided one can construct a dense, continuous class with strong constraint enforcement.
  • The convergence guarantee assumes exact asymptotic minimization of L_{xi,N}; since training uses ADAM with a stochastically sampled loss, a separate estimate of the optimization error would turn Theorem 4.3 into a practical a priori bound.
  • Testing the method with one-hidden-layer ReLU networks and a(t)=ReLU, the architecture the proofs cover, would directly probe whether the proven guarantee holds in practice or whether the numerical success depends on the unproven SoftPlus/t^2 choices.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a mesh-free deep learning method for the obstacle problem based on a first-order system least-squares (FOSLS) reformulation. It rewrites the obstacle problem as the minimization of a least-squares functional L over triples (u, flux, multiplier), enforces boundary and admissibility conditions strongly by representing u = g + d_∂Ω a(v) and γ = div φ + a(η), and analyzes the Monte Carlo sampled losses by Γ-convergence. The main theoretical result, Theorem 4.3, asserts that if parameter sequences asymptotically minimize the sampled regularized functionals, then the induced neural solutions converge in the W-norm to the exact obstacle solution under Assumption 4.1. Section 4.3 verifies Assumption 4.1 for one-hidden-layer ReLU networks with a(t)=ReLU(t), and separately for a piecewise-constant hybrid HNN for η. Numerical experiments in dimensions 10 and 20 and on a slit domain use multi-layer SoftPlus networks and a(t)=t^2.

Significance. If the theoretical analysis were matched by the implementation, the paper would provide a convergence guarantee for a mesh-free obstacle solver with exact constraint enforcement, extending the deep FOSLS framework to variational inequalities. The strong enforcement construction, the introduction of the multiplier through γ, the available code, and the high-dimensional experiments are useful contributions. The principal caveat is that the convergence analysis and the implemented method are not aligned: the theory is verified only for a(t)=ReLU and one-hidden-layer networks, while all experiments use a(t)=t^2 and multi-layer networks. Thus the numerical claims are not currently supported by Theorem 4.3, and several proof details in the Γ-convergence argument need to be supplied.

major comments (4)
  1. [Section 5 vs. §4.3.1 and Assumption 4.1] There is a load-bearing mismatch between the theory and the implementation. Assumption 4.1(2) fixes a Lipschitz continuous function a; §4.3.1 fixes a(t)=ReLU(t) (Eq. (4.11)), and Lemma 4.6 is proved using |a'|=1. However, Section 5 states: 'Across all experiments, we utilized a(t):=t^2'. For a(t)=t^2, the structural hypothesis fails because t^2 is not globally Lipschitz, and the key composition lemma also fails: the gradient error contains a'(w_m)∇(w_m-w)=2w_m∇(w_m-w), which is not controlled in L^2 by H^1 convergence of w_m alone. Consequently Propositions 4.1 and 4.2, and hence Theorem 4.3, do not cover the method actually run in Examples 5.1 and 5.2. The same examples also use 2- and 3-layer SoftPlus networks, whereas the only verification of (H1)-(H2) is for one-hidden-layer networks in Eq. (4.10). The authors should either restrict the experiments to the verified setting or extend t
  2. [Theorem 4.2, lim-inf inequality] The lim-inf part of the Γ-convergence proof is only sketched: after invoking (H2) and Lemma 4.1, it asserts that Gi(Θ_N, ·) → Gi(Θ, ·) a.e. and then says one can 'readily reproduce the arguments presented in [6, Theorem 3.2]'. For G3 and G4 this passage is nontrivial, because it involves products of norm-convergent sequences, e.g. γ_N(u_N−g) and φ_N·∇(u_N−g). One must pass to a common subsequence and use the L^1 domination from Lemma 4.1 before applying dominated convergence. Since this is the hinge of the Γ-convergence theorem, the argument should be written out explicitly rather than deferred.
  3. [Lemma 4.4 / Definition 4.3] The proof of Lemma 4.4 states that H^1(Ω,0,1)=H^1(Ω,1,1). With Definition 4.3 as given, these spaces are not equal: H^1(Ω,0,1) requires w∈L^2(Ω), while H^1(Ω,1,1) only requires d_∂Ω w∈L^2(Ω), a weaker condition near the boundary. The density of C^∞ in the space actually needed, H^1(Ω,0,1), is therefore not justified by the cited argument. Since Proposition 4.1 relies on this density, the authors should either prove the density directly for H^1(Ω,0,1) or give a correct weighted-space reference; if the equality was intended with a different convention, the notation in Definition 4.3 should be corrected and used consistently.
  4. [§4.3.2, Theorem 4.4 and Remark 2] For the HNN_k construction, Theorem 4.4 does not verify hypothesis (H2) as stated. It proves L^2 continuity of Θ↦ηΘ only for Θ0 outside a finite union of subspaces N, and shows that each degenerate Θ∈N has an equivalent nondegenerate Θ'∉N. This does not make the parameter-to-function map continuous on all of B(0,R): sequences approaching a degenerate parameter can converge to a different function, so the lim-inf inequality in Theorem 4.2 is not established for those parameters. A.e. continuity plus reparametrization of individual points is insufficient for the Γ-convergence framework used in the paper. The authors should either modify the admissible parameterization to remove degeneracies continuously, or state the convergence theorem for parameters outside a null set and adapt the Γ-convergence proof accordingly.
minor comments (4)
  1. [Proposition 4.1, last display] In the definition of p_{Θ^m}, the third component is written as div ψ_{Θ^m_ψ} + η_{Θ^m_η}. To belong to W_m it should be div ψ_{Θ^m_ψ} + a(η_{Θ^m_η}), consistent with Eq. (4.2) and with the preceding estimate (4.15). This appears to be a typo, but it matters because the constructed p_{Θ^m} must be admissible.
  2. [Theorem 4.3] The statement is confusing: it first fixes m and R, then concludes by choosing m0, R=R(m0), N0. Clarify that the sequence {Θ_N} is constructed only after m0 and R0 have been selected from Lemma 4.2, so that the Γ-convergence and equi-coercivity arguments apply to the same parameter space.
  3. [Assumption 4.1(2)] There is a typo: 'Lipchitz' should be 'Lipschitz'. More importantly, when the paper says 'we need to fix a Lipschitz continuous function a', it should explicitly remind the reader that this excludes the quadratic function used in Section 5; currently the contradiction is easy to miss.
  4. [Section 5] The text says the experiments 'interchangeably used' the discrete versions of L and J. Since the convergence theory is developed only for L, a sentence explaining the relation between the two discrete functionals and why using JN is expected to preserve the theory would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Γ-convergence argument is carried out in the paper, and the main self-citation is a proof template rather than an unverified load-bearing premise.

full rationale

The paper's claimed derivation is conditional: Theorem 4.3 assumes Assumption 4.1 (density H1 and continuity H2) and then proves, via Γ-convergence and equi-coercivity, that asymptotic minimizers of the sampled functionals converge to p0. The steps are not circular. Coercivity (Prop. 2.1) is imported from Fuehrer [16, Thm. 1] and rewritten under the bijective norm-preserving map (2.9); this is an external, independent result, not a premise of this paper. The Γ-convergence theorem (Thm. 4.2) is stated and proved with the paper's own Lemmas 4.1–4.3; the reference to [6, Thm. 3.2] is for a standard dominated-convergence/lim-inf maneuver whose ingredients (L1 bounds from Lemma 4.1, pointwise a.s. convergence from Lemma 4.3) are supplied in the text, so the self-citation is not load-bearing in the sense of smuggling an unverified premise. Hypotheses (H1) and (H2) are verified in Section 4.3 for one-hidden-layer ReLU networks and for hybrid HNNk networks, with proofs included (Props. 4.1–4.3, Thm. 4.4). No parameter is fitted to a subset of data and then renamed a prediction; the empirical loss L_N is a Monte Carlo surrogate of the same functional L used in the analysis. The reviewer-level concern that the experiments use a(t)=t^2 while the proof fixes a(t)=ReLU (Eq. 4.11) is a genuine applicability gap (correctness risk), not circularity: the paper nowhere defines t^2 as what the theorem covers, and the mismatch does not make Theorem 4.3's conclusion equivalent to its inputs. The numerical examples are demonstrations, not predictions forced by construction. Therefore no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The variables phi, gamma, and eta are auxiliary mathematical variables, and the Lagrange multiplier is standard for obstacle problems. The surrogate distance function is a computational device, not an entity requiring independent falsifiable evidence. The main ledger entries are the structural assumptions and the divergence between the theoretically analyzed network class and the class used in experiments.

free parameters (4)
  • Regularization radius R = chosen in (4.3) and Theorem 4.3
    The convergence theorem requires fixing R>0 in the loss and only guarantees convergence for discrete minimizers inside this ball; its value is not specified or tuned in the experiments.
  • SoftPlus steepness beta = 100
    Used for u and phi activations in all experiments; chosen by hand, not derived from or tuned by the theory.
  • STE parameter c = 0.5
    Used in Example 5.2 to approximate derivatives through Heaviside activations; chosen by hand.
  • Network widths and collocation counts = e.g. 100 or 150 neurons, 20000 or 4000 collocation points
    Hand-chosen for each experiment; not part of the convergence analysis and not justified by a parameter study.
assumptions (7)
  • standard math Equivalence of obstacle problem (2.1) with least-squares problem (2.6), from Fuehrer [16, Theorem 1]
    Prop 2.1 and the characterization p0=(u0, grad u0, -f) are quoted from this external theorem; the paper relies on it without reproving it.
  • domain assumption Omega is bounded and Lipschitz, f in L2, g in H1_0 (Assumption 4.1(1))
    Needed for Poincare, Hardy, and density arguments in Sections 2 and 4; the slit-domain experiment is explicitly outside this assumption.
  • domain assumption A surrogate distance function d_dOmega exists with c1 dist <= d <= c2 dist (Definition 3.1)
    Used to enforce boundary conditions strongly; for complex domains such functions can be difficult to construct and the paper defers to reference [33].
  • ad hoc to paper Activation functions for v and phi are Lipschitz, and eta uses bounded or Lipschitz activations (Assumption 4.1(2))
    Structural assumption used in Lemma 4.1 and the Gamma-convergence proof; the experiments do not verify this for all used networks.
  • ad hoc to paper Approximation hypothesis (H1) and continuity hypothesis (H2) in Assumption 4.1(3)-(4)
    These are assumptions for the general convergence theorem; they are verified only for one-hidden-layer ReLU networks and HNNk, not for the multi-layer SoftPlus networks used in Section 5.
  • ad hoc to paper The auxiliary function a is ReLU in the theory (4.11), but a(t)=t^2 in all experiments
    Lemma 4.6 and the convergence proofs rely on Lipschitz composition properties of ReLU; the paper does not prove that t^2 satisfies the same hypotheses.
  • ad hoc to paper The optimization is assumed to produce a sequence Theta_N with L_{xi,N}(Theta_N) approaching the infimum (Theorem 4.3)
    The main convergence theorem is conditional on asymptotic minimization of the discrete loss; the paper does not prove that ADAM or gradient descent achieves this.

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Cite this review

Pith. "Pith review of A deep first-order system least squares method for the obstacle problem." pith.science (2026). https://pith.science/paper/6CP6QKYW

@misc{pith2026250819412,
  author       = {Pith},
  title        = {Pith review of: A deep first-order system least squares method for the obstacle problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CP6QKYW}},
  note         = {Machine review of arXiv:2508.19412}
}
abstract

We propose a deep learning approach to the obstacle problem inspired by the first-order system least-squares (FOSLS) framework. This method reformulates the problem as a convex minimization task; by simultaneously approximating the solution, gradient, and Lagrange multiplier, our approach provides a flexible, mesh-free alternative that scales efficiently to high-dimensional settings. Key theoretical contributions include the coercivity and local Lipschitz continuity of the proposed least-squares functional, along with convergence guarantees via $\Gamma$-convergence theory under mild regularity assumptions. Numerical experiments in dimensions up to 20 demonstrate the method's robustness and scalability, even on non-Lipschitz domains.

Figures

Figures reproduced from arXiv: 2508.19412 by the authors.

Figure 5
Figure 5. corresponds to [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 5.1
Figure 5.1. Top left: A slice of the computational solution uΘ to example (5.1) in the case d = 10, r0 = 0.7, and R0 = 2. In the computation, we used a learning rate ℓ = 0.001, with 20000 collocation points, 10000 optimization steps, and 65112 degrees of freedom. The bottom left panel exhibits the obstacle and the exact solution as a function of r = |x|. We also report the evolution of the loss function (top right) and the L 2 … view at source ↗
Figure 5.2
Figure 5.2. A slice of the computational solution uΘ (top left), evolution of the loss function (top right), the exact solution and the obstacle as a function of r = ∥x∥2 (bottom left), and evolution of the L 2 -error (bottom right) for (5.1) with d = 20, r0 = 0.9, and R0 = 2. We used a learning rate ℓ = 0.001, with 20000 collocation points, 14000 optimization steps, and 148672 degrees of freedom. performs robustly even on non-… view at source ↗
Figures from the paper (2 more)
Figure 5.3
Figure 5.3. Figure 5.3: Computational solution uΘ (top left), evolution of the loss function (top right), the obstacle function g (bottom left), and the computed contact set η 2 Θ (bottom right). We used a learning rate ℓ = 0.001, with 4000 collocation points, 10000 optimization steps, and …
Figure 5.4
Figure 5.4. Figure 5.4: The computational solution uΘ (black wire-frame), and the obstacle function g (gray), in Example 5.2. [20] J. He, L. Li, J. Xu, and C. Zheng. Relu deep neural networks and linear finite elements. J. Comput. Math., 38(3):502–527, 2020. [21] K. Hornik. Approximation ca…

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