AI weather models may simulate the atmosphere via particle positions in latent space whose updates follow gradient flow on a learned free energy functional rather than conventional physical equations.
A mathematical theory of attention
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Lipschitz continuous transformations F of probability measures w.r.t. Wasserstein distance admit continuous transport maps f(·,μ) such that F(μ) = f(·,μ)_# μ.
Derives forward and backward propagation-of-chaos bounds for finite vs. infinite-context transformers modeled as contextual flow maps, achieving Wasserstein rate n^{-1/d} generally and n^{-1/2} for transformer-like cases.
Lightning self-attention coefficients are coordinates on an algebraic variety obeying Chow-type, low-rank, Veronese-type, and Sylvester-resultant invariants.
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The physics of AI weather models
AI weather models may simulate the atmosphere via particle positions in latent space whose updates follow gradient flow on a learned free energy functional rather than conventional physical equations.
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Continuous transformations of probability measures and their transport representations
Lipschitz continuous transformations F of probability measures w.r.t. Wasserstein distance admit continuous transport maps f(·,μ) such that F(μ) = f(·,μ)_# μ.
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Propagation of Chaos in Contextual Flow Maps
Derives forward and backward propagation-of-chaos bounds for finite vs. infinite-context transformers modeled as contextual flow maps, achieving Wasserstein rate n^{-1/d} generally and n^{-1/2} for transformer-like cases.
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Algebraic Invariants of Lightning Self-Attention
Lightning self-attention coefficients are coordinates on an algebraic variety obeying Chow-type, low-rank, Veronese-type, and Sylvester-resultant invariants.