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A Mathematical Guide to Operator Learning

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arxiv 2312.14688 v1 pith:OZ2TXWJN submitted 2023-12-22 math.NA cs.AIcs.LGcs.NA

classification math.NAcs.AIcs.LGcs.NA
keywords learningoperatorarchitecturesdataexplainguidenetworkneural
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory

    hep-lat 2025-09 conditional novelty 6.0 of 10

    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.

  2. Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning

    cs.LG 2025-06 conditional novelty 4.0 of 10

    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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