Abstract: A method for regression and classification is presented, combining the interpretability of symbolic regression with the efficiency of decision trees, while allowing physical invariances to be guaranteed. The work is motivated by a problem from plasma physics for fusion energy: how does the shaping of the magnetic field affect turbulence in the plasma? Of particular interest is how the field shape could be optimized to reduce the average turbulent flux of heat out of the confinement region. To understand and model this dependence of the heat flux on geometry, regression is performed on a large dataset of direct numerical simulations of turbulence. The underlying physical equations are invariant to translation in the direction along the magnetic field, so the regression should respect this property. This invariance can be guaranteed by applying any translation-invariant reduction to translation-equivariant functions of the geometry. Using many such combinations, a large library of candidate features is defined, and extra nonlinearity is included efficiently by applying decision-tree regression to these features. Sequential feature selection is applied to identify the most important features from the library. Each such feature is an analytic expression involving the magnetic field geometry, providing natural interpretability.
Abstract: We present recent results concerning finite element discretization of PDEs posed on closed surfaces of limited regularity. The first topic pertains to the space semidiscretization with an unfitted method called TraceFEM of the heat equation. Assuming no geometric error, we derive necessary and sufficient conditions for the semidiscrete (Banach-Nečas) inf-sup stability of our method in a robust mesh- dependent norm and discuss its numerous consequences. The second topic deals with the stationary Stokes problem. In this work, we reformulate the Stokes system as a nonsymmetric, indefinite elliptic problem for the velocity-pressure pair at the continuous level. We then take advantage of the new structure to design novel numerical schemes that circumvent the Babuška-Brezzi discrete inf-sup condition. The first topic is joint with L. Bouck, R. H. Nochetto and V. Yushutin, while the second one is joint with R. H. Nochetto
Abstract: Digital Twins (DTs) are adaptive, real-time virtual replicas of physical systems that integrate physics-based models, sensor data, and intelligent decision-making. At their core, DTs can be rigorously framed within PDE–constrained optimization (PDECO).
This talk develops a unified PDECO framework for state estimation and control, employing adjoint-based methods in both deterministic and stochastic settings. To meet the challenges of infinite-dimensional, large-scale optimization, we introduce novel trust-region and augmented Lagrangian algorithms formulated in function spaces.
Beyond these advances, we discuss connections between PDECO and modern machine learning, including how score-based generative models can be interpreted as backward-in-time PDEs. This perspective illustrates how physics-informed modeling and data-driven synthesis can complement one another.
Applications span structural and biomedical systems—from bridges and dams to aneurysm modeling, optimal insulation, electromagnetic cloaking, light bending, and neuromorphic computing—illustrating a pathway toward predictive, adaptive, and trustworthy Digital Twins.
Abstract: Weak-to-strong (W2S) generalization is an intriguing paradigm where a strong, pre-trained student model adapts to a downstream task using pseudo-labels generated by a weaker teacher. Despite the apparent limitations of the weak teacher, W2S fine-tuning often leads the student to outperform the teacher itself. This talk will present two recent theoretical perspectives on why and when W2S succeeds.
First, I will discuss the phenomenon through the lens of low intrinsic dimension and in a variance-dominant regime where fine-tuning often takes place in sufficiently expressive low-dimensional subspaces. This analysis reveals a surprising virtue of discrepancy between strong and weak models' feature representation: while variance is inherited in overlapping subspaces, it is dramatically reduced in subspaces of discrepancy, with explicitly derived characterizations of sample complexity and scaling behavior.
Second, I will examine W2S under spurious correlations, a common challenge when labeled data shaping the teacher and unlabeled data guiding the student differ in group proportions. High-dimensional asymptotic analysis reveals that alignment between group distributions is critical: under group-balanced teachers, minority enrichment improves W2S, while under imbalanced teachers, it harms performance. To address this, a simple confidence-based retraining scheme with generalized cross-entropy can mitigate the pitfalls and consistently strengthen W2S across synthetic and real-world datasets.
Together, these works explain why W2S emerges—via intrinsic dimension and representation discrepancy—and how it is affected by spurious correlations, providing a sharper theoretical foundation and guidance for its future development.
Abstract: Diffusion model is a prevailing Generative AI approach. It uses a score function to characterize a complex data distribution and its evolution toward an easy distribution. This talk will report progress in two different topics, both closely related to the origins of the score function.
The first topic, which will take most time of the talk, will be on a quantification of the generation accuracy of diffusion model. The importance of this problem already led to a rich and substantial literature; however, most existing theoretical investigations assumed that an epsilon-accurate score function has already been oracle-given, and focused on just the inference process of diffusion model. I will instead describe a first quantitative understanding of the actual generative modeling protocol, including both score training (optimization) and inference (sampling). The resulting full error analysis will elucidate (again, but this time theoretically) how to design the training and inference processes for effective generation.
The second topic will no longer be about generative modeling, but sampling instead. The goal is leverage the fact that diffusion model is very good at handling multimodal distributions, and extrapolate it to the holy grail problem of efficient sampling from multimodal density. There, one needs to rethink about how to get the score function, as no more data samples are available and one instead has unnormalized density. A new sampler that is insensitive to metastability, with performance guarantee, and not even requiring continuous density, will be presented.
Abstract: Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a family of new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the p-order algorithm leads to a surprising 2p-order superconvergence, with an error preconstant independent of the number of spatial grids. The proof of superconvergence is based on semiclassical analysis that is of independent interest.
Abstract: The talk presents the first rigorous error analysis of an unfitted finite element method for linear parabolic problems posed on time-dependent domains that may undergo topological changes. The domain evolution is assumed to be smooth away from a critical time, at which the topology may change. To accommodate such transitions in the error analysis, we introduce several structural assumptions on the evolution of the domain in the vicinity of the critical time. These assumptions guarantee a specific control over the variation of a solution norm in time, even across singularities, and form the foundation for the numerical analysis. We demonstrate the applicability of our assumptions with examples of level-set domains undergoing topological transitions and discuss cases where analysis fails. The theoretical error estimate is supported by the results of a numerical experiment. Questions that remain open will be outlined.
Abstract: Continuous data assimilation tackles time-dependent problems with unknown initial conditions by integrating observational data into a nudging term. In this work, we focus on the heat equation with spatially varying conductivity and Neumann boundary conditions, and propose assimilation schemes that utilize non-interpolant observables, diverging from traditional approaches. These generalized nudging techniques are particularly effective for problems that lack regularity beyond the minimal framework. We establish that the spatially discretized nudged solution converges exponentially in time to the true solution, with a convergence rate determined solely by the nudging strategy—independent of the discretization method. Moreover, the long-term discrete error achieves optimality, aligning with known estimates for problems with limited regularity and known initial conditions. We numerically investigate three nudging strategies: Nudging via a conforming finite element subspace; Nudging using piecewise constant functions on the boundary mesh; Nudging based on the mean value of the solution. These strategies are tested on three benchmark problems, including one with Dirac delta forcing and the Kellogg problem characterized by discontinuous conductivity.
Abstract: We consider Chorin's projection method combined with a finite element spatial discretization for the time-dependent incompressible Navier-Stokes equations. The projection method advances the solution in two steps: A prediction step which computes an intermediate velocity field that is generally not divergence-free, and a projection step which enforces (approximate) incompressibility by projecting this velocity onto the (approximately) divergence-free subspace. We establish convergence, up to a subsequence, of the numerical approximations generated by the projection method with finite element spatial discretization to a Leray-Hopf solution of the incompressible Navier-Stokes equations, without any additional regularity assumptions beyond square-integrable initial data and square-integrable forcing. A discrete energy inequality yields a priori estimates, which we combine with a Aubin-Lions type compactness result to prove precompactness of the approximations in $L^2([0,T]\times\dom)$, where $[0,T]$ is the time interval and $\dom$ is the spatial domain. Passing to the limit as the discretization parameters vanish, we obtain a weak solution of the Navier–Stokes equations. A central difficulty is that different a priori bounds are available for the intermediate and projected velocity fields; our compactness argument carefully integrates these estimates to complete the convergence proof. If time permits, I will also discuss how the proof can be adapted to prove convergence of a second-order in time method.
Abstract: In a cryo-electron microscopy experiment, a biomolecular sample is prepared in solution, flash frozen, and imaged. In a processed dataset, each resulting image captures an individual biomolecule in a particular conformation. These images can be averaged to reconstruct the 3D structure of the biomolecule at atomic resolution. With recent advances in microscopes and algorithmic development, the cryo-EM community has focused on the much more ambitious task of "heterogeneity analysis": estimating the probabability distribution of the biomolecule from the images. The "heterogeneity" problem has become extremely popular, with many classical numerical methods to new machine learning approaches. Here, we focus on two aspects of the problem: our work on (1) illuminating a statistical fallacy common in heterogeneity analysis, and (2) a non-parametric maximum likelihood approach which can be used to post-process many existing heterogeneity methods. We highlight theoretical guarantees and numerical simplicity of our approach, and outline the many future opportunities of numerical methods for this inverse problem.
Abstract: Surface Stokes equations have attracted significant recent attention in numerical analysis because approximation of their solutions poses significant obstacles not encountered in the Euclidean context. One of these challenges involves simultaneously enforcing tangentiality and continuity of discrete velocity approximations. Existing finite element methods all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. However, a robust and systematic construction of surface Stokes finite element spaces with nodal degrees of freedom is still missing. In this talk, we introduce a novel approach addressing these challenges by constructing surface finite element spaces with tangential velocity fields based on the classical Taylor-Hood pair. Functions in the discrete spaces are not continuous, but do have conormal-continuity, and the resulting methods do not require ad hoc penalization. We prove stability and optimal-order energy-norm convergence of the method and provide numerical examples illustrating the theory. At the end of the talk, we discuss analogous divergence-free pairs, based on the Scott-Vogelius element. This is joint work with Alan Demlow, Texas A&M.