Deep operator network approximation rates for Lipschitz operators
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it