Kernel Methods are Competitive for Operator Learning
We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator π’^β : π°βπ± are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations Ο(u_i), Ο(v_i) of input/output functions v_i=π’^β (u_i) (i=1,β¦,N), and the measurement operators Ο : π°ββ^n and Ο : π±ββ^m are linear. Writing Ο : β^n βπ° and Ο : β^m βπ± for the optimal recovery maps associated with Ο and Ο, we approximate π’^β with π’Μ =ΟβfΜ βΟ where fΜ is an optimal recovery approximation of f^β :=Οβπ’^β βΟ : β^n ββ^m. We show that, even when using vanilla kernels (e.g., linear or MatΓ©rn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.
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