An optimizer’s view of rare event analysis
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Notes
2025
Temporal Normalizing Flows
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Normalizing flows provide a practical way to learn a probability density and sample from it by pushing forward a simple base distribution (e.g., a standard Gaussian) through an invertible mapping. This post describes a temporal extension—conditioning the flow on time, to model time-dependent densities \(p(t,x)\), with a physics-informed regularization based on a Fokker–Planck equation (FPE).
Data-driven Mori–Zwanzig reduction for cascading failures
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In this note, I revisit an older research direction at the intersection of stochastic dynamical systems, reduced-order modeling, and power-system stability. The motivating question is how to construct effective low-dimensional models for rare but consequential events such as cascading line failures.
Reconstructing PDE solutions
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In this short note, we discuss a data-driven viewpoint for reconstructing solutions of reduced partial differential equations arising from stochastic dynamical systems. A representative setting is the evolution of probability densities associated with a dynamical system of the form
A survey of math behind normalizing flows
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In this short note, we present normalizing flow as an expressive and powerful generative model under which both sampling and density evaluations are efficiently able to be implemented. Possible applications of normalizing flow can be found in solving high-dimensional partial differential equations (PDE).