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arxiv: 2603.06431 · v2 · submitted 2026-03-06 · 🧮 math.NA · cs.LG· cs.NA· stat.ML

Recognition: no theorem link

Certified and accurate computation of function space norms of deep neural networks

Johannes Gr\"undler , Moritz Maibaum , Philipp Petersen
Authors on Pith no claims yet

Pith reviewed 2026-05-15 15:03 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAstat.ML
keywords neural networkscertified computationinterval arithmeticSobolev normsadaptive quadratureLebesgue norms
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The pith

Interval arithmetic on refined boxes yields certified Lebesgue and Sobolev norms for neural networks.

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

The paper develops a method to compute guaranteed lower and upper bounds on integral quantities of neural networks such as L^p and Sobolev norms. It does this by using interval arithmetic to enclose function values and derivatives on axis-aligned boxes, then adaptively refining the boxes where the enclosure is loose and aggregating via quadrature. This matters for applications like physics-informed neural networks where reliable error bounds in function space are needed, since point samples alone cannot guarantee tight bounds on norms. The approach includes a convergence theorem for the adaptive quadrature procedure.

Core claim

By propagating local interval enclosures of function values, Jacobians, and Hessians through adaptive marking and quadrature aggregation, the framework computes certified global bounds on L^p, W^{1,p}, and W^{2,p} norms of deep neural networks, with the enclosures becoming arbitrarily tight under refinement for standard activations.

What carries the argument

Interval arithmetic enclosures of network activations and derivatives on axis-aligned boxes, combined with adaptive refinement and quadrature-based aggregation to obtain global integral bounds.

If this is right

  • Provides certified bounds on PINN interior residuals.
  • The adaptive procedure converges to the true norm value.
  • Extends to computation of other integral quantities beyond norms.
  • Enables rigorous error control for neural network approximations of PDE solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method could extend to certifying other properties like stability or robustness measures of neural networks.
  • Practical implementation might require careful choice of initial box partitions to ensure efficiency.
  • Connections to verified computing in scientific computing could lead to hybrid symbolic-numeric approaches for NN analysis.

Load-bearing premise

The activations and their derivatives must admit computable interval enclosures on axis-aligned boxes that can be made tight through refinement.

What would settle it

A specific neural network where the computed bounds on the L2 norm fail to approach the true value even after extensive adaptive refinement.

Figures

Figures reproduced from arXiv: 2603.06431 by Johannes Gr\"undler, Moritz Maibaum, Philipp Petersen.

Figure 1
Figure 1. Figure 1: Overview of the adaptive refinement strategies. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Boundaries of affine linear pieces for a random ReLU network of width 40 and depth 5. Middle: Adaptive partition after 30 refinement steps of AdaQuad with color indicating local error bounds. The boundaries of affine linear regions are overlaid. Right: Adaptive partition after 40 refinement steps of AdaQuad with color indicating local error bounds. The boundaries of affine linear regions are overlaid… view at source ↗
Figure 3
Figure 3. Figure 3: Mean and 95% confidence interval of the normalized global error for Sobolev norms for 100 untrained tanh networks, refined adaptively, comparing a deep (three hidden layers with 32 neurons) and a wide (one hidden layer with 200 neurons) architecture. Once the adaptive refinement sufficiently resolves localized features, the spatial variance of the error decreases. This results in a more uniform distributio… view at source ↗
Figure 4
Figure 4. Figure 4: Mean and 95% confidence interval of the normalized global error for Sobolev norms of 100 tanh networks, refined with Dörfler marking, comparing a deep (three hidden layers with 32 neurons) and a wide (one hidden layer with 200 neurons) architecture. The neural networks are trained to fit the Gaussian peak (47) function. where r = ∥x∥2, ϵ = 0.5 and h(t) = e −1/t for t > 0 and zero otherwise. We train two ar… view at source ↗
Figure 5
Figure 5. Figure 5: Mean and 95% confidence interval of the normalized global error for the L p norm for 100 ReLU networks, comparing a deep and wide architecture. The trained neural networks approximate the one-dimensional Gaussian-peak function (47). 4.5.4 Bounding the energy norm for elliptic PINNs We analyze the interior residual bound calculated by bounding the energy norm as in Proposi￾tion 3.13 and its theoretical conv… view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional disk experiment: adaptive evaluation of the normalized global error [PITH_FULL_IMAGE:figures/full_fig_p037_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two-dimensional disk experiment: Convergence rate of the normalized error for the [PITH_FULL_IMAGE:figures/full_fig_p038_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Heatmaps of local errors for the Sobolev norm for the deep architecture (left) and [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: PINN (49) experiment: convergence of the global error for the energy norm and local error reduction (Remark 4.2). 45 [PITH_FULL_IMAGE:figures/full_fig_p045_9.png] view at source ↗
read the original abstract

Neural network methods for PDEs require reliable error control in function space norms. However, trained neural networks can typically only be probed at a finite number of point values. Without strong assumptions, point evaluations alone do not provide enough information to derive tight deterministic and guaranteed bounds on function space norms. In this work, we move beyond a purely black-box setting and exploit the neural network structure directly. We present a framework for the certified and accurate computation of integral quantities of neural networks, including Lebesgue and Sobolev norms, by combining interval arithmetic enclosures on axis-aligned boxes with adaptive marking/refinement and quadrature-based aggregation. On each box, we compute guaranteed lower and upper bounds for function values and derivatives, and propagate these local certificates to global lower and upper bounds for the target integrals. Our analysis provides a general convergence theorem for such certified adaptive quadrature procedures and instantiates it for function values, Jacobians, and Hessians, yielding certified computation of $L^p$, $W^{1,p}$, and $W^{2,p}$ norms. We further show how these ingredients lead to practical certified bounds for PINN interior residuals. Numerical experiments illustrate the accuracy and practical behavior of the proposed methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims a framework for certified computation of integral quantities of neural networks, including L^p, W^{1,p}, and W^{2,p} norms, by using interval arithmetic enclosures on axis-aligned boxes combined with adaptive marking/refinement and quadrature aggregation. It states a general convergence theorem for such certified adaptive quadrature procedures and instantiates it for function values, Jacobians, and Hessians, with further application to certified bounds on PINN interior residuals. Numerical experiments are used to illustrate accuracy and practical behavior.

Significance. If the convergence theorem holds with enclosures that tighten sufficiently for standard activations including ReLU, the work would offer a practical method for obtaining guaranteed bounds on function-space norms of neural networks without relying solely on pointwise probes. This addresses a key need in neural PDE solvers for rigorous error control beyond black-box settings, and the use of interval methods with adaptive quadrature provides a concrete, implementable approach with potential for reproducibility via the described enclosures and quadrature rules.

major comments (2)
  1. [abstract and convergence theorem statement] The general convergence theorem (instantiated in the abstract for Jacobians and Hessians) assumes that local interval enclosures for the integrand and its derivatives can be driven to arbitrary accuracy by box refinement. However, for ReLU networks the first and second derivatives are piecewise constant with jumps across hyperplanes; standard interval arithmetic on axis-aligned boxes incurs wrapping and dependency errors that do not necessarily vanish uniformly when kink surfaces intersect the boxes. No separate argument is supplied showing that enclosure widths still contract at a rate sufficient for global bounds to converge to the true integral.
  2. [framework description] The weakest assumption stated—that network activations and derivatives admit computable interval enclosures on axis-aligned boxes—is not accompanied by a precise characterization of the network class (e.g., depth, width, or activation restrictions) for which the enclosures remain tight enough to guarantee practical convergence of the adaptive procedure.
minor comments (2)
  1. [quadrature-based aggregation] Clarify in the introduction or methods section whether the quadrature rules are chosen to be compatible with the interval enclosures (e.g., exact integration of enclosure bounds) or if additional overestimation is introduced.
  2. [numerical experiments] The numerical experiments section would benefit from explicit reporting of enclosure widths versus refinement level for at least one ReLU example to allow readers to assess the practical contraction rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below and will revise the manuscript to incorporate clarifications where needed.

read point-by-point responses

  1. Referee: [abstract and convergence theorem statement] The general convergence theorem (instantiated in the abstract for Jacobians and Hessians) assumes that local interval enclosures for the integrand and its derivatives can be driven to arbitrary accuracy by box refinement. However, for ReLU networks the first and second derivatives are piecewise constant with jumps across hyperplanes; standard interval arithmetic on axis-aligned boxes incurs wrapping and dependency errors that do not necessarily vanish uniformly when kink surfaces intersect the boxes. No separate argument is supplied showing that enclosure widths still contract at a rate sufficient for global bounds to converge to the true integral.

    Authors: We thank the referee for highlighting this subtlety. The general convergence theorem (Theorem 3.1) is stated under the assumption that enclosure diameters tend to zero with box diameters, which holds pointwise away from discontinuities. For ReLU networks the derivatives are discontinuous across hyperplanes of measure zero. The adaptive procedure refines boxes intersecting these surfaces, and the total volume of such boxes can be driven to zero while the integrand remains bounded (by construction of the network). The contribution of these boxes to the integral error is therefore controlled by their vanishing measure times a uniform bound on the integrand, ensuring the global certified bounds converge to the true value. We will add a dedicated remark after the theorem statement clarifying this argument for the non-smooth case without changing the theorem itself. revision: yes

  2. Referee: [framework description] The weakest assumption stated—that network activations and derivatives admit computable interval enclosures on axis-aligned boxes—is not accompanied by a precise characterization of the network class (e.g., depth, width, or activation restrictions) for which the enclosures remain tight enough to guarantee practical convergence of the adaptive procedure.

    Authors: We agree that a sharper characterization improves clarity. The framework applies to finite-depth, finite-width feedforward networks whose activations are continuous and admit computable interval enclosures together with their derivatives on compact intervals (ReLU, sigmoid, tanh, softplus, etc.). Enclosures are obtained recursively layer by layer via standard interval arithmetic; the resulting overestimation is reduced by the adaptive subdivision. We will revise the assumptions paragraph in Section 2 to state this network class explicitly and note that the method requires only that min/max of each activation and its derivative are computable on intervals, which covers all activations used in the numerical experiments. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard interval arithmetic and adaptive quadrature

full rationale

The paper derives certified L^p, W^{1,p}, and W^{2,p} norm bounds by combining interval enclosures on axis-aligned boxes, adaptive marking/refinement, and quadrature aggregation. The general convergence theorem for certified adaptive quadrature is instantiated directly from the properties of interval arithmetic (tightening under refinement) and standard quadrature rules, without reducing any target quantity to a fitted parameter, self-defined input, or load-bearing self-citation. The method is self-contained against external interval libraries and does not rename known results or smuggle ansatzes via prior work by the same authors. No equation or step in the abstract or described framework exhibits a reduction by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the existence of computable interval enclosures for the chosen activations and their derivatives, plus standard properties of interval arithmetic and adaptive quadrature convergence.

axioms (2)
  • domain assumption Interval arithmetic produces rigorous enclosures for the network and its derivatives on axis-aligned boxes
    Invoked when the paper states that guaranteed lower and upper bounds for function values and derivatives are computed on each box.
  • standard math Adaptive marking and refinement combined with quadrature yields convergent global bounds
    The general convergence theorem for certified adaptive quadrature procedures.

pith-pipeline@v0.9.0 · 5518 in / 1365 out tokens · 40924 ms · 2026-05-15T15:03:56.883352+00:00 · methodology

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