Transformer residual layers are approximated as an explicit Euler scheme for a controlled hidden-state flow whose mean-field limit is a first-order transport control problem with Pontryagin terminal condition given by the softmax residual.
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Training installs a depth-dependent spectral gradient and low-rank bottleneck in LLM residual streams whose amplification or suppression of graph communities is predicted by local operator type.
Regularization of Fourier multipliers yields L^∞ stability for wave propagation and compact operator inversion in inverse problems.
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A First-Order Mean Field Control Analysis of Transformer Layers under Cross-Entropy Training
Transformer residual layers are approximated as an explicit Euler scheme for a controlled hidden-state flow whose mean-field limit is a first-order transport control problem with Pontryagin terminal condition given by the softmax residual.
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Dynamics of the Transformer Residual Stream: Coupling Spectral Geometry to Network Topology
Training installs a depth-dependent spectral gradient and low-rank bottleneck in LLM residual streams whose amplification or suppression of graph communities is predicted by local operator type.
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On $L^\infty$ stability for wave propagation and for linear inverse problems
Regularization of Fourier multipliers yields L^∞ stability for wave propagation and compact operator inversion in inverse problems.