An operator learning perspective on parameter-to-observable maps
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1.
California Institute of Technology
Abstract
Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven surrogates that map between function spaces. However, instead of full-field measurements, often the available data are only finite-dimensional parametrizations of model inputs or finite observables of model outputs. Building on Fourier Neural Operators, this paper introduces the Fourier Neural Mappings (FNMs) framework that is able to accommodate such finite-dimensional vector inputs or outputs. The paper develops universal approximation theorems for the method. Moreover, in many applications the underlying parameter-to-observable (PtO) map is defined implicitly through an infinite-dimensional operator, such as the solution operator of a partial differential equation. A natural question is whether it is more data-efficient to learn the PtO map end-to-end or first learn the solution operator and subsequently compute the observable from the full-field solution. A theoretical analysis of Bayesian nonparametric regression of linear functionals, which is of independent interest, suggests that the end-to-end approach can actually have worse sample complexity. Extending beyond the theory, numerical results for the FNM approximation of three nonlinear PtO maps demonstrate the benefits of the operator learning perspective that this paper adopts.
Copyright and License
© 2024 American Institute of Mathematical Sciences.
Acknowledgement
The first author is supported by the high-performance computing platform of Peking University. The second author acknowledges support from the National Science Foundation Graduate Research Fellowship Program under award number DGE-1745301 and from the Amazon/Caltech AI4Science Fellowship, and partial support from the Air Force Office of Scientific Research under MURI award number FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation). The third author is supported by the Department of Energy Computational Science Graduate Fellowship under award number DE-SC00211. The second and third authors are also grateful for partial support from the Department of Defense Vannevar Bush Faculty Fellowship held by Andrew M. Stuart under Office of Naval Research award number N00014-22-1-2790.
The computations presented in this paper were partially conducted on the Resnick High Performance Computing Center, a facility supported by the Resnick Sustainability Institute at the California Institute of Technology. The authors thank Kaushik Bhattacharya for useful discussions about learning functionals, Andrew Stuart for helpful remarks about the universal approximation theory, and Zachary Morrow for providing the code for the advection–diffusion equation solver. The authors are also grateful for the helpful feedback from two anonymous referees.
Data Availability
Links to datasets and all code used to produce the numerical results and figures in this paper are available at
Additional details
Related works
- Is new version of
- Discussion Paper: arXiv:2402.06031 (arXiv)
- Is supplemented by
- Dataset: https://github.com/nickhnelsen/fourier-neural-mappings (URL)
Funding
- Peking University
- National Science Foundation
- DGE-1745301
- California Institute of Technology
- Amazon/Caltech AI4Science Fellowship
- United States Air Force Office of Scientific Research
- FA9550-20-1-0358
- United States Department of Energy
- DE-SC00211
- United States Department of Defense
- Vannevar Bush Faculty Fellowship
- Office of Naval Research
- N00014-22-1-2790
Dates
- Available
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2024-08Early access