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arxiv: 2606.09820 · v2 · pith:L2JUOU5Enew · submitted 2026-06-08 · 🧮 math.FA · cs.LG· math.PR· q-fin.MF· stat.ML

Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

Philipp Schmocker , Josef Teichmann This is my paper

Pith reviewed 2026-06-30 11:30 UTC · model grok-4.3

classification 🧮 math.FA cs.LGmath.PRq-fin.MFstat.ML
keywords universal approximationdifferentiable mapsinfinite-dimensional manifoldsNachbin theoremfunctional neural networkspath signaturesnon-anticipative functionalsweighted approximation
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The pith

A weighted Nachbin theorem establishes universal approximation for differentiable maps on infinite-dimensional manifolds including their derivatives.

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

The paper generalizes the universal approximation theorem for functional input neural networks so that they approximate not only maps from weighted manifolds to Banach spaces but also the derivatives of those maps. The key step is a new weighted version of Nachbin's density theorem that works simultaneously for the function values and the derivatives. This removes the usual restriction to compact sets and yields approximation results for non-anticipative functionals together with their horizontal and vertical derivatives. One concrete application is that linear functions of the signature can approximate path-space functionals along with their directional derivatives. A reader would care because many models in stochastic analysis and infinite-dimensional settings require control over both the output and its sensitivity to changes in the input.

Core claim

By proving a weighted Nachbin theorem, the authors show that the function algebras generated by functional neural networks are dense in the space of differentiable maps on weighted manifolds, where the topology controls both the maps and their derivatives. This produces universal approximation theorems for differentiable maps that hold on infinite-dimensional weighted manifolds rather than only on compact sets, and it implies approximation results for non-anticipative functionals including horizontal and vertical derivatives. Linear functions of the signature are shown to approximate path-space functionals together with their directional derivatives.

What carries the argument

The weighted Nachbin theorem, which supplies density of suitable subalgebras in the space of differentiable functions on a weighted manifold with respect to a topology that includes derivative control.

If this is right

  • Functional neural networks approximate non-anticipative functionals together with their horizontal and vertical derivatives.
  • Linear functions of the signature approximate path-space functionals and their directional derivatives.
  • The approximation result applies simultaneously to a map and its derivatives on infinite-dimensional weighted manifolds.
  • The density holds in a topology that controls derivatives, extending beyond the usual compact-set setting.

Where Pith is reading between the lines

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

  • If the weighted manifold condition holds for a given class of activations, the same density argument could be checked for higher-order derivatives.
  • The signature approximation result suggests testing the method on concrete path-dependent problems arising in stochastic processes.
  • The framework may connect to existing signature-based models by supplying derivative control that those models currently lack.

Load-bearing premise

The input space must admit a weighted manifold structure that makes the Nachbin-type density result hold simultaneously for the maps and their derivatives under the chosen activation functions.

What would settle it

A concrete counterexample would be a specific weighted manifold together with a differentiable map such that no sequence of functional neural networks approximates both the map and its derivative uniformly to arbitrary precision.

Figures

Figures reproduced from arXiv: 2606.09820 by Josef Teichmann, Philipp Schmocker.

Figure 1
Figure 1. Figure 1: A FNN φ : M → Y with additive family A, activation function ρ ∈ C k (R), linear readout L ⊆ Y , and N = 3 number of neurons. Remark 4.5. Definition 4.4 extends the notion of classical neural networks between Euclidean spaces. Indeed, let φ : R d → R m be a classical neural network of the form (4.2) R d ∋ x 7→ φ(x) = W ρ(Ax + b) = X N n=1 ynρ [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Learning f1 defined in (7.1) by path-NN φ ∈ FN ρ,ρe Λ α,r T,R (label FNN) and linear function of the signature P 0≤|I|≤NSig aI ⟨eI , S(Xb )t, eI ⟩ (label Sig). In (a), the weighted mean squared error (7.3) is evaluated on the training set in each epoch (con￾tinuous line) as well as on the test after every 200-th epoch (dots). In (b)–(d), three sam￾ples x(m) of the test set are shown together with f1(·, x(m… view at source ↗
Figure 3
Figure 3. Figure 3: Learning f2 defined in (7.2) by path-NN φ ∈ FN ρ,ρe Λ α,r T,R (label FNN) and linear function of the signature P 0≤|I|≤NSig aI ⟨eI , S(Xb )t, eI ⟩ (label Sig). In (a), the weighted mean squared error (7.3) is evaluated on the training set in each epoch (con￾tinuous line) as well as on the test after every 200-th epoch (dots). In (b)–(d), three sam￾ples x(m) of the test set are shown together with f2(·, x(m… view at source ↗
read the original abstract

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

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 / 1 minor

Summary. The manuscript generalizes the universal approximation theorem for functional neural networks (FNNs) mapping from possibly infinite-dimensional weighted manifolds to Banach spaces. It proves a weighted Nachbin theorem to establish density results that simultaneously approximate differentiable maps and their derivatives, extending beyond compact sets. Applications include approximation of non-anticipative functionals (with horizontal and vertical derivatives) and linear functionals of the signature for path-space functionals including directional derivatives.

Significance. If the weighted Nachbin theorem and its applications hold, the result would extend classical UATs to differentiable maps on non-compact infinite-dimensional spaces, providing a tool for approximation theory in functional analysis with potential relevance to stochastic analysis and signature methods. The explicit inclusion of derivative approximation and the weighted setting are the main novelties.

major comments (2)
  1. [Abstract] The central claim rests on the existence of a weighted manifold structure compatible with the Nachbin density result for differentiable maps (abstract and the paragraph introducing the weighted Nachbin theorem). Without an explicit definition of this structure, the precise conditions on the weight function, and verification that the activation functions satisfy the required density simultaneously for the map and derivatives, the proof cannot be assessed for correctness.
  2. The transition from the weighted Nachbin theorem to the approximation of non-anticipative functionals (including horizontal and vertical derivatives) and to signature approximations requires additional technical steps that are not verifiable from the provided abstract; these steps appear load-bearing for the applications but lack outlined proofs or references to prior results.
minor comments (1)
  1. Clarify the precise statement of the weighted Nachbin theorem (e.g., the topology on the space of differentiable maps and the form of the weight) in the introduction or dedicated section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify points from the manuscript. We address each major comment below, noting that the referee's concerns appear to stem from the abstract alone; the full text contains the requested definitions, conditions, and proof outlines.

read point-by-point responses

  1. Referee: [Abstract] The central claim rests on the existence of a weighted manifold structure compatible with the Nachbin density result for differentiable maps (abstract and the paragraph introducing the weighted Nachbin theorem). Without an explicit definition of this structure, the precise conditions on the weight function, and verification that the activation functions satisfy the required density simultaneously for the map and derivatives, the proof cannot be assessed for correctness.

    Authors: The full manuscript explicitly defines the weighted manifold structure immediately after introducing the weighted Nachbin theorem, specifies the precise conditions on the weight function required for compatibility with the density result, and verifies in the proof that the chosen activation functions achieve simultaneous density for both the maps and their derivatives. These elements are standardly located in the body rather than the abstract, which is a high-level summary. revision: no

  2. Referee: [—] The transition from the weighted Nachbin theorem to the approximation of non-anticipative functionals (including horizontal and vertical derivatives) and to signature approximations requires additional technical steps that are not verifiable from the provided abstract; these steps appear load-bearing for the applications but lack outlined proofs or references to prior results.

    Authors: The full manuscript details the technical transitions in the dedicated applications section. The steps from the weighted Nachbin theorem to the approximation results for non-anticipative functionals (including horizontal and vertical derivatives) and signature-based approximations are outlined with explicit references to prior results on these topics, and the proofs are provided in the text. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper claims to prove a weighted Nachbin theorem that directly yields a UAT for differentiable maps on weighted infinite-dimensional manifolds, including derivative approximation. No equations, definitions, or derivation steps are exhibited that reduce any claimed result to its inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract and description frame the work as an independent proof extending prior results without self-referential reductions, making the derivation self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence proof in functional analysis; no numerical fitting or new postulated objects are described in the abstract.

axioms (1)
  • standard math Standard results from functional analysis on Banach spaces, manifolds, and weighted topologies
    The proof invokes known properties of infinite-dimensional manifolds and weighted approximation to establish the Nachbin-type density.

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