Blog

The big-data scaling recipe applies unevenly across scientific machine learning: it works for equilibrium tasks like AlphaFold but hits a structural ceiling at per-trajectory prediction of chaotic systems.

LLM inference nondeterminism is rarely a hardware artifact: dynamic batching switches kernels, the kernel switch reorders floating-point summations, and the resulting drift in logits silently turns RLVR training off-policy.

What is the adjoint of the Koopman operator on a stochastic Markov process, when is the operator normal, and why does the singular value decomposition become the natural object the moment normality fails?

Building the finite-dimensional spectral theorem from the basis-free adjoint, normality, and Gram–Schmidt induction on eigenvectors. Part I of a three-part series, with a separate Operator SVD capstone.

The compact case between finite dimension and general bounded operators: the variational principle as the eigenvalue source, compactness itself as the discretizer that forces the eigenvalues to decay to zero. Part II of a three-part series, with a separate Operator SVD capstone.

When a bounded self-adjoint operator has no eigenvectors at all, the eigenbasis is replaced by a projection-valued measure built from continuous functional calculus. Part III of a three-part series, with a separate Operator SVD capstone.

The SVD as the spectral theorem applied to L*L via the polar form L = U|L|: why it is the optimal low-rank object, what shape it takes at each spectral-theorem rung, and where it stops existing.

The library default for TOA solar radiation was a different physical quantity than GraphCast's training data. How I diagnosed and reconstructed the correct one.

Materialized intermediates, fit-once transforms, executor per stage, quarantined dependencies: four design decisions in a single-node ETL for terabyte-scale weather data, with the alternatives I rejected.