This is the schedule for the Nonlinear PDE seminar in Texas A&M University.
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All times showed below are local Texas time (CDT/CST). Please consult WorldTimeBuddy for your local time.
Please contact Minh-Binh Tran at minhbinh_at_tamu_dot_edu or Xin Liu at xliu23_at_tamu_dot_edu for any question.
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.
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