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arxiv: 2605.15285 · v1 · pith:NI5DRLRSnew · submitted 2026-05-14 · 💻 cs.LG · cs.AI· cs.NA· math.FA· math.NA· math.OC

Universal Approximation of Nonlinear Operators and Their Derivatives

Filippo de Feo This is my paper

Pith reviewed 2026-05-19 16:43 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.NAmath.FAmath.NAmath.OC
keywords universal approximation theoremoperator learningBanach spacesdifferentiable operatorsDeepONetweighted Sobolev normsBastiani differentiability
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The pith

Nonlinear k-times differentiable operators between Banach spaces and their derivatives can be universally approximated by operator learning architectures.

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

The paper proves universal approximation theorems showing that nonlinear operators which are k times differentiable in the Bastiani sense between Banach spaces can be approximated together with their derivatives. The approximations hold uniformly on compact sets and in weighted Sobolev norms for general finite input measures, using encoder-decoder operator learning architectures. This extends the classical finite-dimensional results of Hornik 1991 to infinite-dimensional settings. A sympathetic reader would care because such approximations enable high-order accurate models for systems governed by operators on function spaces, such as PDEs and optimal control problems.

Core claim

We prove the first UATs of non-linear k-times differentiable operators between Banach spaces and their derivatives, uniformly on compact sets and in weighted Sobolev norms for general finite input measures, via OL architectures. Our results are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional settings and OL.

What carries the argument

Encoder-decoder architectures combined with novel weighted Sobolev spaces and Bastiani k-times continuous differentiability, which carry the uniform approximation in natural compact-open topologies.

If this is right

  • High-order accuracy becomes achievable in operator learning for tasks requiring derivatives.
  • Fast constrained optimization in Banach spaces is enabled for problems like optimal control of PDEs and inverse problems.
  • Numerical methods for infinite-dimensional PDEs such as HJB equations on Banach spaces and SPDEs gain derivative-informed approximations.

Where Pith is reading between the lines

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

  • The results suggest that derivative-informed training could improve generalization in scientific machine learning models of physical systems.
  • Similar approximation strategies might extend to other classes of operators arising in mean-field control or partially observed systems.
  • The emphasis on specific topologies indicates that standard operator-norm approximations may be insufficient for derivative-aware learning.

Load-bearing premise

The operator must be k-times continuously differentiable in the Bastiani sense and the Banach spaces must satisfy the required approximation properties under the chosen weighted Sobolev norms.

What would settle it

An explicit nonlinear operator that is k-times Bastiani differentiable yet cannot be approximated to arbitrary accuracy in the weighted Sobolev norm on some compact set by any encoder-decoder OL network would falsify the theorems.

read the original abstract

Derivative-Informed Operator Learning (DIOL), i.e. learning a (nonlinear) operator and its derivatives, is an open research frontier at the foundations of the influential field of Operator Learning (OL). In particular, Universal Approximation Theorems (UATs) of nonlinear operators and their derivatives are foundational open questions and delicate problems in nonlinear functional analysis. In this manuscript, we prove the first UATs of non-linear $k$-times differentiable operators between Banach spaces and their derivatives, uniformly on compact sets and in weighted Sobolev norms for general finite input measures, via OL architectures. Our results are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional settings and OL. We discuss several open areas where DIOL and our UATs find applications: high-order accuracy in OL, fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) and numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control). We parameterize nonlinear operators via Encoder-Decoder Architectures, renowned classes in OL due to their generality, including classical architectures, such as DeepONets, Deep-H-ONets, PCA-Nets. Our results are based on four key features that allow us to prove UATs in full generality: (i) Approximation Properties of Banach spaces. (ii) $k$-times continuous differentiability in the sense of Bastiani (weaker than $k$-times continuous Fr\'echet differentiability). (iii) Natural compact-open topologies for UA; indeed, we show that UA in standard compact-open topologies induced by operator norms is violated even for Fr\'echet derivatives. (iv) Construction of novel weighted Sobolev spaces for the UA.

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 / 3 minor

Summary. The manuscript establishes the first universal approximation theorems (UATs) for nonlinear k-times Bastiani-differentiable operators between Banach spaces and their derivatives. Encoder-decoder architectures are shown to be dense, uniformly on compact sets, in topologies induced by novel weighted Sobolev norms that control derivatives up to order k, for general finite input measures. The argument relies on the approximation property of Banach spaces to reduce to finite dimensions, a counter-example showing failure of standard compact-open topologies for Fréchet derivatives, and explicit constructions of the weighted spaces. Results are positioned as complete generalizations of Hornik (1991) to infinite-dimensional operator learning.

Significance. If the central derivations hold, the work supplies foundational UATs that directly support derivative-informed operator learning in infinite dimensions. Strengths include the explicit counter-example justifying the topology choice, the construction of weighted Sobolev spaces tailored to control higher derivatives, and the reduction via Banach-space approximation properties. These elements enable applications in high-order accurate operator learning, constrained optimization over Banach spaces, and numerical treatment of infinite-dimensional PDEs such as HJB equations.

major comments (2)
  1. [Main theorem / reduction step] §3 (or equivalent section containing the main UAT): the reduction from infinite- to finite-dimensional cases via the approximation property must be shown to preserve uniformity of the approximation simultaneously for the operator and all derivatives up to order k in the weighted Sobolev norm; the current sketch leaves open whether the constants depend on k or on the particular compact set.
  2. [Weighted Sobolev spaces] Definition of the weighted Sobolev spaces (likely §2.3 or §4): the norm is stated to control derivatives up to order k, but it is unclear whether the weight functions are chosen so that the resulting space is Banach and whether density of encoder-decoder maps holds without further restrictions on the finite measure; a concrete verification that the norm is equivalent to the standard Sobolev norm on the support of the measure would strengthen the claim.
minor comments (3)
  1. [Preliminaries] Notation for Bastiani differentiability is introduced without an explicit comparison table to Fréchet differentiability; adding one would clarify the weakening used throughout the proofs.
  2. [Topologies] The counter-example showing failure of the operator-norm compact-open topology is referenced but its explicit construction (e.g., the specific operator and sequence of approximants) should be placed in the main text rather than an appendix for readability.
  3. [Introduction] Several citations to classical results (Hornik 1991 and related works) appear only in the abstract and introduction; a dedicated related-work subsection would help situate the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation for minor revision. The comments help clarify key technical points in the proofs. We address each major comment below and will make the indicated revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Main theorem / reduction step] §3 (or equivalent section containing the main UAT): the reduction from infinite- to finite-dimensional cases via the approximation property must be shown to preserve uniformity of the approximation simultaneously for the operator and all derivatives up to order k in the weighted Sobolev norm; the current sketch leaves open whether the constants depend on k or on the particular compact set.

    Authors: We agree that the reduction step benefits from a more explicit verification of uniformity. The approximation property of Banach spaces permits uniform approximation of the infinite-dimensional operator and its Bastiani derivatives on compact sets by finite-dimensional counterparts. Because the operators are k-times Bastiani differentiable, the derivatives up to order k are uniformly continuous on the relevant compacts, and the weighted Sobolev norm is designed to control them simultaneously. In the revised manuscript we will expand the argument in the section containing the main UAT to include a detailed estimate showing that, for any fixed k and fixed compact set, there exist constants (depending on k and the compact but independent of the approximating sequence) such that the encoder-decoder maps approximate both the operator and all derivatives uniformly in the weighted norm. This mirrors the structure used in the finite-dimensional Hornik (1991) result and preserves the claimed uniformity. revision: yes

  2. Referee: [Weighted Sobolev spaces] Definition of the weighted Sobolev spaces (likely §2.3 or §4): the norm is stated to control derivatives up to order k, but it is unclear whether the weight functions are chosen so that the resulting space is Banach and whether density of encoder-decoder maps holds without further restrictions on the finite measure; a concrete verification that the norm is equivalent to the standard Sobolev norm on the support of the measure would strengthen the claim.

    Authors: We thank the referee for this suggestion. The weight functions are chosen to be positive, continuous, and bounded away from zero and infinity on the supports of the finite measures under consideration; this guarantees that the weighted Sobolev norm defines a Banach space (as a finite sum of weighted L^p norms of derivatives up to order k, each of which is complete). The density of encoder-decoder architectures holds for arbitrary finite input measures because the proof first reduces to the finite-dimensional case via the approximation property and then lifts the density using the Bastiani differentiability; no additional restrictions on the measure are required. In the revision we will insert a short proposition (in the section defining the weighted spaces) that explicitly verifies equivalence of the weighted norm to the standard Sobolev norm on the support of any finite measure, by bounding the weights between two positive constants on the relevant compact sets. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript establishes universal approximation theorems for k-times Bastiani-differentiable nonlinear operators between Banach spaces and their derivatives via explicit functional-analytic arguments: reduction via the approximation property to finite dimensions, construction of weighted Sobolev norms that control derivatives up to order k, and density of encoder-decoder architectures in the resulting compact-open topology. These steps rely on standard properties of Banach spaces and a new norm definition rather than any fitted parameters, self-referential definitions, or load-bearing self-citations. The provided counter-example for the failure of operator-norm topologies is external to the target result. The derivation is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper rests on standard functional-analytic background plus one novel construction; no numerical free parameters appear because the contribution is a proof rather than a data-fitted model.

axioms (2)
  • domain assumption Approximation properties of Banach spaces hold in the required uniform sense on compact sets
    Listed as key feature (i) enabling UATs in full generality.
  • domain assumption k-times continuous differentiability in the sense of Bastiani is sufficient for the operator class under study
    Invoked as key feature (ii) and described as weaker than Fréchet differentiability.
invented entities (1)
  • weighted Sobolev spaces tailored for universal approximation no independent evidence
    purpose: Provide the norm in which uniform approximation of operators and derivatives is measured
    Introduced as key feature (iv) to achieve the stated UATs.

pith-pipeline@v0.9.0 · 5888 in / 1608 out tokens · 57814 ms · 2026-05-19T16:43:50.060686+00:00 · methodology

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