Title: Control, Optimal Transport and Neural Differential Equations in Supervised Learning
Abstract: We study the fundamental computational problem of approximating optimal transport (OT) equations using neural differential equations (Neural ODEs). More specifically, we develop a novel framework for approximating unbalanced optimal transport (UOT) in the continuum using Neural ODEs. By generalizing a discrete UOT problem with Pearson divergence, we constructively design vector fields for Neural ODEs that converge to the true UOT dynamics, thereby advancing the mathematical foundations of computational transport and machine learning. To this end, we design a numerical scheme inspired by the Sinkhorn algorithm to solve the corresponding minimization problem and rigorously prove its convergence, providing explicit error estimates. From the obtained numerical solutions, we derive vector fields defining the transport dynamics and construct the corresponding transport equation.
Finally, from the numerically obtained transport equation, we construct a neural differential equation whose flow converges to the true transport dynamics in an appropriate limiting regime.
This is joint work with M.-B. Tran.
Title:
Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry 2
Abstract:
We present AlphaGeometry 2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with support for non-constructive problems, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry 2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that enables effective communication between search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry 2 to 84% for all geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry 2 was also part of the system that achieved silver-medal standard at IMO 2024. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.
Title: Entropy-driven control of the continuity equation for normalizing flows
Abstract: Normalizing flows are generative models that transform a simple (typically Gaussian) reference probability distribution into a complex target distribution through a sequence of smooth, invertible maps. Within the continuous-time framework of neural ODEs, the objective can be recast as a control problem for the continuity equation: we seek a time-dependent vector field that drives the final-time distribution arbitrarily close to the target with respect to relative entropy.
Under a tail-compatibility assumption, we establish approximate controllability for this system. The proof is constructive and combines a reverse Pinsker inequality with a piecewise-constant-in-time control scheme that yields explicit bounds on the required number of switches. The result sheds light on the reachable set of the continuity equation in relative entropy and links classical control theory with modern flow-based generative modeling.
Title: Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms
Abstract: In this talk we consider Schrödinger PDEs, posed on a Riemannian manifold M, with bilinear control. We propose a new method to prove the global approximate controllability. Contrarily to previous ones, it works in arbitrarily small times and does not require a discrete spectrum. We also present the main ideas of the proof. Our approach consists in controlling separately the radial and the angular parts of the wavefunction, thanks to the control of the group of (compactly supported and isotopic to the identity) diffeomorphisms of M and the control of phases. We develop this approach for two examples of Schrödinger equations, posed on tori and euclidean spaces of arbitrary dimensions, for which the small-time control of phases was recently proved. We prove that it implies the small-time control of flows of vector fields, with Lie bracket techniques. Combining this property with the simplicity of the group of (compactly supported and isotopic to the identity) diffeomorphisms proved by Thurston, we obtain the control of such group. The small-time control of the radial part then follows from the transitivity of the group action of diffeomorphisms on positive normalized densities, proved by Moser.
This talk is based on the article [1], in collaboration with Karine Beauchard (ENS Rennes, France).
[1] K. Beauchard, E. Pozzoli; Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms. To appear in Ann. Inst. Henri Poincaré C Anal. Non Linéaire (2025) arXiv:2410.02383
Title: The L2 theory for compressible Euler equations and inviscid limit from Navier-Stokes equations
Abstract: Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data.
In this talk, I will first discuss our recent L2 stability theory for compressible Euler equations using the method of relative entropy, with several collaborators. This theory is the second major stability theory for hyperbolic conservation laws, after the L1 theory in 1990s. As an application, we proved all BV solutions must satisfy the Bounded Variation Condition proposed earlier by Bressan and Lewicka as a sufficient condition for uniqueness, hence showed the uniqueness of BV solution.
Then I will introduce the recent result on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations. This is a famous open problem after Bressan-Bianchini’s seminal vanishing artificial viscosity limit result for BV solutions of hyperbolic conservation laws. This is a join work with Kang and Vasseur.
Title:
Stability of Type-I Blowup without Spectral Analysis: Theory, Numerics, and Computer-assisted Proofs for Nonlinear PDEs
Abstract:
I will present a unified framework for establishing type-I singularity formation in nonlinear PDEs without relying on spectral analysis, which is crucial for complex dynamics like Navier-Stokes that calls for computer-assisted proofs. The approach aims at establishing stability around a blowup profile, and combines dynamic rescaling with novel modulation parameters that control translation, rotation, and scaling instabilities, allowing us to impose vanishing conditions that eliminate all unstable and neutral modes. This leads to a purely energy-based stability argument and captures the log correction to the blowup rate naturally through modulation dynamics. This formulation naturally suggests a numerical algorithm for tracking singularity formation. We apply this method to the nonlinear heat and complex Ginzburg–Landau equations, and then to the open problem of the 3D Keller–Segel system with logistic damping, where crucially, there are no explicit profiles, and blowup is established with finite codimensional stability. I will also briefly discuss recent progress in computing such solutions, via methods we developed, including high-precision PINNs and Kolmogorov–Arnold Networks.
Title: Free Boundary Regularity and Convergence of Tumor Growth Models
Abstract: In tumor growth models, two primary approaches are commonly used. The first, described by Porous Medium type equations, models the tumor cells as a distribution evolving over space. The second, based on Hele-Shaw type flows, focuses on the evolution of the domain occupied by the cells. These two models are connected through the incompressible limit. In this talk, I will first present results showing that for Hele-Shaw type flows with source and advection terms, flat free boundaries are Lipschitz continuous. I will then discuss the convergence of free boundaries in the incompressible limit. As an outcome, we provide an upper bound on the Hausdorff dimension of free boundaries and show that the limiting free boundary has finite $(d-1)$-dimensional Hausdorff measure. These are joint works with Inwon Kim and Jiajun Tong.
Title: Nested Stochastic Algorithm for Generalized Sinkhorn Distance-Regularized Distributionally Robust Optimization
Abstract: Distributionally robust optimization (DRO) is a powerful technique to train robust models against data distribution shift. This paper aims to solve regularized nonconvex DRO problems, where the uncertainty set is modeled by a so-called generalized Sinkhorn distance and the loss function is nonconvex and possibly unbounded. Such a distance allows to model uncertainty of distributions with different probability supports and divergence functions. For this class of regularized DRO problems, we derive a novel dual formulation taking the form of nested stochastic optimization, where the dual variable depends on the data sample. To solve the dual problem, we provide theoretical evidence to design a nested stochastic gradient descent (SGD) algorithm, which leverages stochastic approximation to estimate the nested stochastic gradients. We study the convergence rate of nested SGD and establish polynomial iteration and sample complexities that are independent of the data size and parameter dimension, indicating its potential for solving large-scale DRO problems. We conduct numerical experiments to demonstrate the efficiency and robustness of the proposed algorithm.
Title: Non-potential mean-field games à la Benamou-Brenier
Abstract: Mean-field games (MFG) theory is a mathematical framework for studying large systems of agents who play differential games. In the PDE form, MFG reduces to a Hamilton-Jacobi equation coupled with a continuity or Kolmogorov-Fokker-Planck equation. Theoretical analysis and computational methods for these systems are challenging due to the absence of strong regularizing mechanisms and coupling between two nonlinear PDE.
One approach that proved successful from both theoretical and computational perspectives is the variational approach, which interprets the PDE system as KKT conditions for suitable convex energy. MFG systems that admit such representations are called potential systems and are closely related to the dynamic formulation of the optimal transportation problem due to Benamou-Brenier. Unfortunately, not all MFG systems are potential systems, limiting the scope of their applications.
I will present a new approach to tackle non-potential systems by providing a suitable interpretation of the Benamou-Brenier approach in terms of monotone inclusions. In particular, I will present advances on the discrete level and numerical analysis and discuss prospects for the PDE analysis.
Title: Control and optimization for Neural Partial Differential Equations in Supervised Learning
Abstract: In supervised learning, the primary objective is to transport initial data toward target data through the layers of a neural network. We propose a novel perspective: the supervised learning task for convolutional neural networks (CNNs) can be interpreted as a control problem for partial differential equations (PDEs). We propose a dual system formulation for the control and optimization problem associated with parabolic PDEs. We also provide a theoretical proof showing that the control and optimization problem for parabolic PDEs admits minimizers. Finally, we investigate the control problem associated with hyperbolic PDEs and prove the existence of solutions for a corresponding approximated control problem.
Title: On admissibility criteria for the compressible Euler equations
Abstract: In the past years, results based on a technique called convex integration have drawn lots of interest within the community of mathematical fluid mechanics. Among other fascinating results, this technique allows to prove existence of infinitely many solutions for the multi-dimensional compressible Euler equations. All these solutions satisfy the energy inequality which is commonly used in the literature to identify physically relevant solutions. On the other hand, intuitively at least some of the infinitely many solutions still seem to be non-physical. For this reason one has studied additional admissibility criteria like the maximal energy dissipation criterion or the least action criterion -- to no avail: such criteria do not select the solution which is expected to be the physical one.
In this talk we give an overview on the aforementioned non-uniqueness results, and we explain why maximal dissipation as well as the least action criterion fail to single out the solution which is presumably the physical solution.
Title: Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation
Abstract: In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach based on the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations. This is joint work with Jonas Lührmann.
Title: AOT algorithm for a system of equations arising in meteorology
Abstract: Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model, while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model. We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.
Title: Viscosity and the formation of shocks
Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. In a nutshell, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. The proof involves matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. This is joint work with Sanchit Chaturvedi and Cole Graham.