Stochastic processes and sampling
Reflected Brownian motion, Markov chains, rare events, Monte Carlo, and high dimensional sampling.
01 · Research
Structure makes scaling possible.
I study when more data, model capacity, optimization effort, or inference computation can be converted into predictable improvements. The common language is mathematics: differential equations, stochastic processes, control, geometry, and randomization.
Why do some models scale while others stall?
Scaling fails when approximation, optimization, statistical uncertainty, or simulation error becomes the hidden bottleneck. I aim to identify that bottleneck mathematically and redesign the algorithm so that added resources improve the quantity we actually care about.
What should be preserved?
Physical laws, operator geometry, stochastic dynamics, and algebraic identities are not auxiliary information. They determine the representation, metric, and estimator. Preserving them often turns an unstable black box into a controllable computational method.
How should theory meet computation?
The objective is not a bound in isolation. A useful theory should identify the right algorithm, predict its scaling behavior, and survive high dimensional experiments. This loop connects my work in machine learning, numerical analysis, and probability.
Trading computation for reliability after training
Can a fixed generative model become more accurate simply by spending more computation at inference time? I approach this through sequential Monte Carlo, importance sampling, stochastic control, data assimilation, and reward twisting. The goal is to make additional computation reduce error in a measurable and theoretically controlled way.
Scientific inference as a second scaling axis
For PDE and inverse problems, a learned approximation can be calibrated through simulation without retraining. The resulting defect equation turns model error into a new computational target and gives an end to end relation between extra simulation and improved accuracy.
Learning functions and operators with the right notion of error
Scientific learning problems are infinite dimensional. Their optimal algorithms depend on the geometry of function spaces, regularity, multilevel structure, and the downstream quantity of interest. My work develops minimax theory and algorithms for PDE solution learning and operator learning.
Encoding physics into architectures
PDE Net learns differential operators as constrained convolution filters. Neural ODE viewpoints connect deep architectures to numerical discretization and optimal control. These connections provide interpretable representations and expose the computational structure needed for analysis.
Uncertainty quantification and debiasing
Learned surrogates are useful only when their error can be detected and corrected. I combine approximation with Monte Carlo, control variates, and orthogonal bootstrap ideas to obtain statistically efficient scientific estimators.
Geometry that survives width and depth
Standard Euclidean smoothness often grows with model size and cannot explain learning rate transfer. I study steepest descent under matrix operator norms and depth aware function space metrics to design optimizers with stable scaling laws.
Reliable randomized numerical algorithms
Randomization accelerates matrix computation, but accuracy alone is not enough. My work asks when randomized solvers are backward stable, how sketching interacts with preconditioning, and how spectral structure controls convergence.
Making mathematical structure searchable
AI systems are most useful for mathematics when the problem is represented in a form that supports exact search, verification, and compositional reuse. I study how apparently analytic or probabilistic arguments can be converted into algebraic objects and correction circuits.
AI assisted discovery with proof responsibility
The signed basic adjoint relationship problem illustrates the workflow: use AI to explore transformations and counterexamples, isolate an invariant algebraic mechanism, and then turn it into a human verifiable theorem with a precisely stated boundary of validity.
Foundations
Mathematics beneath the applications
The applications vary, but the toolkit is intentionally coherent.
PDEs and function spaces
Sobolev and reproducing kernel spaces, operator approximation, multilevel structure, and regularity.
Optimization and linear algebra
Geometric optimization, eigenvalue methods, randomized solvers, preconditioning, and numerical stability.