Research interests
I am mostly interested in Statistical Learning Theory and applying it to obtain mathematical guarantees (convergence rates, error bounds…) for Deep Learning-based solutions of PDEs arising from the study of rare events, such as computation of equilibrium distributions, first exit times or committor functions. For problems of this kind, it is almost always necessary to exploit the extra mathematical structure in order to obtain tractable algorithms.
Below is a brief overview of the kind of problems that have kept me busy throughout my PhD years.
1. Physics-informed neural networks for rare-event PDEs
One of my main research directions concerns physics-informed neural networks (PINNs) for solving PDEs that arise in the analysis of rare events and metastable stochastic dynamics. These include boundary value problems associated with exit times, transition probabilities, and related quantities. Such PDEs are often high-dimensional and may exhibit sharp boundary layers or stiffness, which makes them challenging for classical numerical schemes.
In this direction, I am interested not only in designing PINN architectures and training schemes for these problems, but also in using tools from statistical learning theory to derive error estimates and generalization guarantees. The goal is to understand how the sample complexity and optimization difficulty depend on domain geometry, solution regularity, and the choice of network architecture and training procedure.
Keywords: physics-informed neural networks, rare events, high-dimensional PDEs, generalization bounds.
2. Operator learning for committor functions
In a closely related direction, I have been investigating neural operator learning approaches to learning committor functions, which encode the probability that a stochastic system transitions from one metastable set to another. Committor functions are the central object of study in transition path theory and rare-event analysis, but they are typically difficult (read: intractable) to compute in complex, high-dimensional systems.
The operator learning approach views the mapping from the (infinite-dimensional) parameter space (e.g., the potential landscape) to the associated committor function as an operator to be learned from relevant input-output pairs. I am interested in developing operator-learning architectures that are tailored to this structure, and in establishing theoretical guarantees—for example, sample-complexity bounds and rates of convergence—that quantify when and why these methods can reliably approximate committors in realistic settings.
Keywords: operator learning, committor function, transition path theory, sampling.
3. Fast rates under low-noise conditions for deep networks
In a much more classical flavor, I’ve also been interested in rates of convergence for classification with deep neural networks (DNNs) under (Tsybakov) low-noise conditions, which encode how “hard” a classification problem is. In supervised learning, such conditions are known to yield fast rates (faster than the standard $n^{-1/2}$ rate) for certain estimators, but the extent to which these phenomena carry over to modern deep neural networks is not yet fully understood.
I have thus been looking into how these low-noise conditions (perhaps coupled with further distributional assumptions) could lead to improved statistical guarantees for modern DNNs trained by Empirical Risk Minimization (ERM). The aim is to understand how these low-noise conditions intertwine with DNN architecture choices and training procedures to yield regimes where deep models can provably achieve sharper excess-risk bounds than those predicted by the classical “worst-case” theory.
Keywords: statistical learning theory, Tsybakov low-noise condition, fast rates, deep neural networks, excess risk bounds.