| Abstract: |
| Operator learning aims to approximate mappings between infinite dimensional function spaces, while practical implementations rely on finite dimensional encoder decoder representations that often induce resolution dependent hypothesis classes and obscure the underlying operator being learned. We overcome this ambiguity by using a kernel based framework that identifies a resolution independent class of limiting operators associated with encoder decoder architectures. Starting from matrix valued kernel learning at a fixed finite resolution, we show that the encoder decoder structure admits an isometric lifting to an operator valued kernel formulation on function spaces. As input and output resolutions increase, encoder induced kernels converge to a resolution independent limiting kernel that characterizes the intrinsic operator class. Building on this limiting kernel perspective, we analyze stochastic gradient descent (SGD) for operator learning in encoder decoder settings. The framework applies to broad classes of kernels and encoder decoder constructions, including radial and dot product kernels; spectral truncation based encoders such as Fourier, polynomial, and wavelet based encoders; empirical principal component encoders; and kernel interpolation based encoding schemes. This is a joint work with Jiaqi Yang and Prof. Dingxuan Zhou. |
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