Scaling laws for optimization and inference
We study the geometry, complexity, and resource tradeoffs that determine when neural optimizers and inference algorithms scale predictably.
01 · Research
Structure makes scaling possible.
We identify the mathematical bottlenecks that prevent added data, model capacity, optimization effort, or inference computation from becoming reliable improvement.
Generalization, interpolation, and reliability
Average risk can conceal important failure modes. Our work uses high dimensional statistics and probability to understand generalization, interpolation, and rare severe errors.
Sequential Monte Carlo and stochastic simulation
We design particle methods, importance sampling schemes, and stochastic control formulations that convert additional computation into measurable accuracy gains.
Stochastic networks and rare events
Reflected diffusions, basic adjoint relationships, large deviations, and rare events provide a mathematical language for understanding stability and tail behavior.
Randomized numerical linear algebra
We ask when sketching, preconditioning, and randomized refinement are not only fast but also stable and accurate in finite precision.
PDE computation and uncertainty quantification
Approximation theory, numerical analysis, and stochastic simulation guide algorithms for PDE learning, operator learning, inverse problems, and uncertainty quantification.
Scientific machine learning with correction
Learned models become more trustworthy when physical structure and simulation reveal, quantify, and correct their defects.
AI assisted mathematical discovery
We use AI to explore transformations and counterexamples, then isolate precise mathematical mechanisms and produce human verifiable arguments.