Artificial intelligence is, at its core, applied mathematics. These math applications stands at the intersection of pure math and computer science, diving into the theoretical foundations of modern AI 📐. Explore complex research on gradient descent optimization, high-dimensional linear algebra, probabilistic reasoning, and the calculus of neural networks. From understanding the topology of deep learning models to the statistical theories explaining how massive language models generalize data, discover the mathematical proofs and equations that make artificial intelligence possible.
A lighting artist on an animated film wants a lamp that glows from the inside, shaped like a knotted fractal that a shader programmer wrote in forty lines of code. The renderer can draw that shape easily. It cannot light…
Sampling SDF Volumetric Emitters in Monte Carlo Rendering Read More »
Every time you eliminate a vertex from a graph, its neighbours collapse into a dense little cluster where none may have existed before. Do that thousands of times, as any direct solver for a large sparse system must, and the…
CAST Speeds Up Approximate Cholesky on Graphs Read More »
An oncologist wants to know how a patient tumour will spread over the coming months, and how the nutrients feeding it will move as it grows. The honest answer today involves a supercomputer and a long wait. Simulating how a…
A Neural Operator That Solves Free Boundary Problems Read More »
The idea that batch size shapes generalization dates back at least to Keskar et al. (2017), who showed empirically that large-batch training reliably finds sharper minima and those minima tend to generalize worse. The explanation became common wisdom: stochastic noise…
Why Batch Size Changes What Your Neural Network Learns Read More »
Expected Improvement has a lot going for it. It has a closed-form expression, it naturally trades off exploration and exploitation, and decades of practitioners have found that it just works. Jones, Schonlau, and Welch popularized it in 1998, and it…
When Expected Improvement Falls Short and What EIC Does About It Read More »
Two mathematicians have delivered the first fully rigorous probabilistic derivation of the total QSSA for the Michaelis-Menten enzyme model, settling a long-standing gap in the mathematical foundations of systems biology.
Two mathematicians from the UAE and Uzbekistan have proven that whether a p-adic spin model behaves predictably or descends into complete chaos depends entirely on the arithmetic of a single prime number — a result that opens entirely new territory…
Chaos in the p-adic Ising Model — When Prime Numbers Decide Phase Transitions Read More »
Sen Na at Georgia Tech and Michael Mahoney at UC Berkeley prove that a sketched, adaptive Stochastic SQP method achieves asymptotic normality for constrained nonlinear stochastic optimization — the first online estimator that handles equality constraints without ever computing a…
A team from Carnegie Mellon University flipped conventional wisdom on its head — proving that agents with different behavior policies don’t just tolerate each other’s differences, they actively benefit from them. And a novel importance-averaging scheme eliminates the last remaining…
The Blessing of Heterogeneity in Federated Q-Learning: Linear Speedup and Beyond Read More »
A team from Peking University and Georgia Tech has built a formal theory explaining why the most widely used GNN approach to multi-node tasks breaks down — and proved that a simple but principled labeling strategy solves the problem completely.
Improving Graph Neural Networks on Multi-node Tasks with the Labeling Trick Read More »