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A Mathematical Guide to Operator Learning
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Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively. We also give advice on how to create and manage training data and conduct optimization. We offer intuition behind the various neural network architectures employed in operator learning by motivating them from the point-of-view of numerical linear algebra.
Forward citations
Cited by 2 Pith papers
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Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory
A matrix-free neural preconditioner learns to map gauge configurations to modified configurations whose Dirac operators approximate the inverse, halving CG iterations and transferring across lattice sizes.
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Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
A practical recipe to convert common neural architectures into discretization-agnostic neural operators, validated by Navier-Stokes experiments showing cross-resolution generalization of FNO-style models.
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