Deep operator network approximation rates for Lipschitz operators
Bibliographic record
Abstract
We establish a framework for universality and expression rate bounds for a class of neural Deep Operator Networks (DONs) emulating Lipschitz (or Hölder) continuous maps [Formula: see text] between (subsets of) separable Hilbert spaces [Formula: see text], [Formula: see text]. The DON architecture considered uses linear encoders [Formula: see text] and decoders [Formula: see text] via (biorthogonal) Riesz bases of [Formula: see text], [Formula: see text], and an approximator network of an infinite-dimensional, parametric coordinate map that is Lipschitz continuous on the sequence space [Formula: see text]. Unlike previous works [L. Herrmann, C. Schwab and J. Zech, Neural and spectral operator surrogates: Construction and expression rate bounds, Adv. Comput. Math. 50(4) (2024) 72; C. Marcati and C. Schwab, Exponential convergence of deep operator networks for elliptic partial differential equations, SIAM J. Numer. Anal. 61(3) (2023) 1513–1545] which required for example [Formula: see text] to be holomorphic, the present expression rate results require mere Lipschitz (or Hölder) continuity of [Formula: see text]. Key in the proof of the present expression rate bounds is the use of either superexpressive activations (e.g., [Z. Shen, H. Yang and S. Zhang, Neural network approximation: Three hidden layers are enough, Neural Netw. 141 (2021) 160–173; Z. Shen, H. Yang and S. Zhang, Deep network approximation: Achieving arbitrary accuracy with fixed number of neurons, J. Mach. Learn. Res. 23(276) (2022) 1–60; D. Yarotsky, Elementary superexpressive activations, in Proc. 38th Int. Conf. Machine Learning (PMLR, 2021), pp. 11932–11940] and the references there) which are inspired by the Kolmogorov superposition theorem ([A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114 (1957) 953–956] or [G. G. Lorentz, Approximation of Functions (Holt, Rinehart and Winston, New York–Chicago, IL–Toronto, ON, 1966), Chap. 11] for a comprehensive exposition), or of nonstandard NN architectures with standard (ReLU) activations as recently proposed in [Z. Shen, H. Yang and S. Zhang, Deep network approximation: Achieving arbitrary accuracy with fixed number of neurons, J. Mach. Learn. Res. 23(276) (2022) 1–60; S. Zhang, Z. Shen and H. Yang, Neural network architecture beyond width and depth, in Advances in Neural Information Processing Systems, Vol. 35 (Curran Associates, 2022), pp. 5669–5681]. We illustrate the abstract results by approximation rate bounds for emulation of (a) solution operators for parametric elliptic variational inequalities and (b) Lipschitz maps of Hilbert–Schmidt operators.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.036 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.003 | 0.006 |
| Research integrity | 0.002 | 0.007 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".