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arxiv: 2404.03099 · v1 · pith:YBBNUV4Gnew · submitted 2024-04-03 · 💻 cs.LG · cs.AI· cs.CE· cs.IT· math.IT· stat.ML

Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networks

Leonardo Ferreira Guilhoto , Paris Perdikaris This is my paper

Pith reviewed 2026-05-24 02:18 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CEcs.ITmath.ITstat.ML
keywords neural operatorsBayesian optimizationepistemic uncertaintyfunction spacescomposite optimizationoperator learningsequential decision making
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The pith

NEON uses one operator network to match deep ensembles in composite Bayesian optimization over function spaces while using orders of magnitude fewer parameters.

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

The paper presents NEON as an architecture that produces predictions together with epistemic uncertainty from a single operator network backbone. It applies this to composite Bayesian optimization, where one seeks to maximize an unknown composition f = g ∘ h and h maps inputs to elements of a function space. Experiments on toy problems and real-world tasks show that the resulting acquisition strategy reaches state-of-the-art performance. A sympathetic reader cares because the approach removes the need to train and store large ensembles when the objective involves functional intermediates.

Core claim

NEON is an architecture for generating predictions with uncertainty using a single operator network backbone, which presents orders of magnitude less trainable parameters than deep ensembles of comparable performance. When applied to the problem of composite Bayesian optimization of f = g ∘ h, where h : X → C(𝒴, ℝ^{d_s}) is an unknown map outputting elements of a function space and g is a known cheap functional, NEON achieves state-of-the-art performance on toy and real-world scenarios.

What carries the argument

NEON (Neural Epistemic Operator Networks), a single operator network backbone that supplies epistemic uncertainty estimates to guide acquisition in composite Bayesian optimization over function spaces.

If this is right

  • Composite Bayesian optimization over functional outputs becomes feasible with far lower memory and training cost.
  • Operator learning models can supply the uncertainty needed for sequential decision making without maintaining multiple independent networks.
  • The same backbone can be reused across multiple composite problems that share the same functional output space.
  • Real-time or resource-constrained applications of function-space optimization become practical.

Where Pith is reading between the lines

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

  • NEON-style single-backbone uncertainty could be tested on other sequential tasks that already use operator networks, such as control of PDE-governed systems.
  • If the uncertainty quality generalizes, similar single-network designs might replace ensembles in related operator-learning settings that currently rely on them for calibration.
  • The method invites direct comparison against other cheap uncertainty mechanisms, such as last-layer Laplace approximations, inside the same composite BO loop.

Load-bearing premise

A single operator network backbone can produce epistemic uncertainty estimates of quality comparable to deep ensembles for the purpose of guiding composite Bayesian optimization.

What would settle it

A controlled composite Bayesian optimization benchmark in which NEON-guided search reaches demonstrably worse final values than an ensemble baseline of matched predictive accuracy.

Figures

Figures reproduced from arXiv: 2404.03099 by Leonardo Ferreira Guilhoto, Paris Perdikaris.

Figure 1
Figure 1. Figure 1: Example of h(u) ∈ C([0, 221]2 , R 2 ) for the the Cell Towers problem. The input u ∈ R 30 encodes transmission parameters of 15 cell towers, which are used to produce the function seen above, where signal intensity and interference are plotted, respectively. This information is the used to compute a score f(u) = g(h(u)) ∈ R which evaluates the quality of cellular service in the region. By using operator co… view at source ↗
Figure 2
Figure 2. Figure 2: Diagrams for the architectures used in this paper. The NEON architecture (top) combines the deterministic [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagrams representing the two decoders used in the experiments considered in this paper. On the left, the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental results for the Environment Model (left) and Brusselator PDE (right) problems. In both cases [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experimental results for the Optical Interferometer (left) and Cell Towers (right) problems. For the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Full experimental results for the Environmental Modeling (left) and Brusselator PDE (right) problems. The [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Best result obtained by NEON for the Optical Interferometer problem. Here we plot the 16 components of [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Best result obtained by NEON for the Cell Towers problem. Here we plot the signal strength and interference [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Full experimental results for the Optical Interferometer (left) and Cell Towers (right) problems. The dashed [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Average results among 5 trials comparing parallel acquisition functions using [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
read the original abstract

Operator learning is a rising field of scientific computing where inputs or outputs of a machine learning model are functions defined in infinite-dimensional spaces. In this paper, we introduce NEON (Neural Epistemic Operator Networks), an architecture for generating predictions with uncertainty using a single operator network backbone, which presents orders of magnitude less trainable parameters than deep ensembles of comparable performance. We showcase the utility of this method for sequential decision-making by examining the problem of composite Bayesian Optimization (BO), where we aim to optimize a function $f=g\circ h$, where $h:X\to C(\mathcal{Y},\mathbb{R}^{d_s})$ is an unknown map which outputs elements of a function space, and $g: C(\mathcal{Y},\mathbb{R}^{d_s})\to \mathbb{R}$ is a known and cheap-to-compute functional. By comparing our approach to other state-of-the-art methods on toy and real world scenarios, we demonstrate that NEON achieves state-of-the-art performance while requiring orders of magnitude less trainable parameters.

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

0 major / 2 minor

Summary. The manuscript introduces NEON (Neural Epistemic Operator Networks), an operator-learning architecture that produces epistemic uncertainty estimates from a single network backbone rather than an ensemble. The method is applied to composite Bayesian optimization of the form f = g ∘ h, where h maps to a function space and g is a known, cheap functional; experiments on toy problems and real-world tasks are reported to show state-of-the-art optimization performance while using orders of magnitude fewer trainable parameters than comparable deep ensembles.

Significance. If the experimental results hold, the work supplies a concrete, parameter-efficient mechanism for epistemic uncertainty in infinite-dimensional operator learning that directly supports sequential decision-making. The evaluation measures optimization regret rather than isolated predictive metrics, and the architecture description supplies an explicit route to uncertainty that is shown to be competitive with ensembles; these elements strengthen the central claim.

minor comments (2)
  1. The notation for the composite objective (h : X → C(Y, R^{d_s})) is introduced in the abstract but would benefit from an explicit reminder in the first paragraph of §3 when the BO acquisition functions are defined.
  2. Figure 2 caption states that NEON uses 'a single backbone'; a one-sentence clarification of how the epistemic head is attached without increasing the parameter count relative to a deterministic operator network would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough review, positive assessment of the significance of NEON for uncertainty-aware operator learning in composite Bayesian optimization, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims rest on introducing the NEON architecture for epistemic uncertainty via a single operator-network backbone and then validating its utility for composite Bayesian optimization through direct empirical comparisons against SOTA baselines on toy and real-world tasks. These comparisons measure optimization performance (not merely internal predictive metrics) and are independent of any self-referential definitions, parameter fits renamed as predictions, or load-bearing self-citations. No equations or architectural choices in the provided description reduce by construction to the target results; the derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5722 in / 920 out tokens · 20300 ms · 2026-05-24T02:18:59.225521+00:00 · methodology

discussion (0)

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