In 1998, Steve Smale proposed his 18th mathematical problem for the 21st century, calling for a deeper understanding of the fundamental limitations of intelligence. Rapid advances in artificial intelligence over the past decade have lent renewed urgency to this challenge, underscoring the need for principled theoretical frameworks that characterize the capabilities and limitations of modern learning systems.

The structural differences between human intelligence and contemporary artificial intelligence models shed light on central issues in the development of trustworthy AI, including robustness, uncertainty quantification, and computational efficiency. In this talk, I examine these questions through the lens of modern statistical theory, with a particular emphasis on recent advances in false discovery rate control and their role in establishing mathematical guarantees for trustworthy artificial intelligence.

In this talk, we propose a new physics-informed kernel approach to solve general nonhomogeneous PDEs. Specifically, physical priors (typically encoded through PDE operator information) are incorporated into a kernel ridge regression formulation, and a regularization-based approach is employed to construct the operator learner. The method proposes a closed-form solution that is independent of the parameters of the associated PDE. From the perspective of regularization theory, the resulting estimator induces a well-defined operator that links the input and output function spaces. Consequently, it effectively shifts from a PDE solver to an operator-based solver, achieving outcomes comparable to those obtained with neural operator learning. In contrast to standard supervised learning frameworks, the proposed approach enables systematic extrapolation beyond the regimes represented in the observations without any training. Finally, a full error analysis is conducted that provides convergence rates using adaptive regularization parameters for operator-based solver.

Artificial intelligence (AI) based Molecular Sciences have begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all these AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization. Molecular structures and their interactions are represented by high-order topological and algebraic models (including Rips complex, Alpha complex, Neighborhood complex, Dowker complex, Hom-complex, Tor-algebra, Rhombille tiling, etc). Mathematical invariants (from persistent homology, Ricci curvature, persistent spectral, Analytic torsion, algebraic variety, etc) are used as molecular descriptors for learning models. Further, we develop geometric and topological deep learning models that can systematically incorporate molecular high-order, multiscale, and periodic information, and use them for analysing molecular data from chemistry, biology, and materials. 

The parameter space of a neural network is often used as a proxy for the space of functions it represents, yet this correspondence is typically non-injective: distinct parameter configurations may realize the same function due to underlying symmetries. While such functional equivalence has been well studied in classical architectures, its role in modern models remains far less understood. In this talk, I will present our recent progress on the symmetry structure of modern neural architectures such as Transformers and Mixture-of-Experts. I will then discuss applications of these findings to equivariant metanetwork design and linear mode connectivity.

Distribution regression seeks to estimate the conditional distribution of a multivariate response given a continuous covariate. This approach offers a more complete characterization of dependence than traditional regression methods. Classical nonparametric techniques often assume that the conditional distribution has a well-defined density, an assumption that fails in many real-world settings. These include cases where data contain discrete elements or lie on complex low-dimensional structures within high-dimensional spaces. In this work, we establish minimax convergence rates for distribution regression under nonparametric assumptions, focusing on scenarios where both covariates and responses lie on low-dimensional manifolds. We derive lower bounds that capture the inherent difficulty of the problem and propose a new hybrid estimator that combines adversarial learning with simultaneous least squares to attain matching upper bounds. Our results reveal how the smoothness of the conditional distribution and the geometry of the underlying manifolds together determine the estimation accuracy.