REVIEW 6 cited by
Data Complexity Estimates for Operator Learning
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
Data Complexity Estimates for Operator Learning
abstract
Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain incomplete. The present work develops theory to study the data complexity of operator learning, complementing existing research on the parametric complexity. We investigate the fundamental question: How many input/output samples are needed in operator learning to achieve a desired accuracy $\epsilon$? This question is addressed from the point of view of $n$-widths, and this work makes two key contributions. The first contribution is to derive lower bounds on $n$-widths for general classes of Lipschitz and Fr\'echet differentiable operators. These bounds rigorously demonstrate a ``curse of data-complexity'', revealing that learning on such general classes requires a sample size exponential in the inverse of the desired accuracy $\epsilon$. The second contribution of this work is to show that ``parametric efficiency'' implies ``data efficiency''; using the Fourier neural operator (FNO) as a case study, we show rigorously that on a narrower class of operators, efficiently approximated by FNO in terms of the number of tunable parameters, efficient operator learning is attainable in data complexity as well. Specifically, we show that if only an algebraically increasing number of tunable parameters is needed to reach a desired approximation accuracy, then an algebraically bounded number of data samples is also sufficient to achieve the same accuracy.
Forward citations
Cited by 6 Pith papers
-
Transpose-free linear algebra
Establishes non-identifiability results and query lower bounds showing transpose-free matvec access provides limited information for core linear algebra tasks.
-
Is Zero-Shot Super-Resolution Possible in Operator Learning?
Zero-shot super-resolution is information-theoretically impossible for some simple operators but possible under Hölder smoothness of outputs, accompanied by generalization bounds.
-
From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators
FNOs achieve polynomial sample complexity for learning time-T solution operators of dissipative evolution equations when those operators admit stable spectral discretizations, with rates depending on smoothness, dimen...
-
Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces
Introduces variation spaces for nonlinear operators and derives dimension-independent approximation bounds of order N^{-1/2} plus encoding errors for encoder-decoder two-layer networks, yielding algebraic rates under ...
-
Fourier Neural Operators for Time-Periodic Quantum Systems: Learning Floquet Hamiltonians, Observable Dynamics, and Operator Growth
FNOs learn three maps for time-periodic spin chains (Floquet Hamiltonian, local observables, operator growth) with high accuracy, zero-shot transfer across time grids and driving frequencies, and extrapolation beyond ...
-
A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
A survey of operator learning theory showing holomorphy gives fast sample-complexity rates, general smoothness gives a polylogarithmic barrier, and FNO-approximable classes cap out at n^{-1/2}.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.