Kernel Methods are Competitive for Operator Learning
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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 𝒢^† : 𝒰→𝒱 are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations ϕ(u_i), φ(v_i) of input/output functions v_i=𝒢^†(u_i) (i=1,…,N), and the measurement operators ϕ : 𝒰→ℝ^n and φ : 𝒱→ℝ^m are linear. Writing ψ : ℝ^n →𝒰 and χ : ℝ^m →𝒱 for the optimal recovery maps associated with ϕ and φ, we approximate 𝒢^† with 𝒢̅=χ∘f̅∘ϕ where f̅ is an optimal recovery approximation of f^†:=φ∘𝒢^†∘ψ : ℝ^n →ℝ^m. We show that, even when using vanilla kernels (e.g., linear or Matérn), 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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