2025 Simons Collaboration on Wave Turbulence Annual Meeting

The annual meeting of the Collaboration on Wave Turbulence was held at the Simons Foundation on December 4–5. It was preceded by a three-day workshop at the Courant Institute that focused on closely related topics and highlighted recent contributions by collaborators and postdoctoral researchers. The workshop featured updates spanning a broad range of research directions, from fundamental theoretical advances to innovative applications in geophysical and other physical systems. These presentations prompted in-depth discussions and lively exchanges, providing a natural and productive lead-in to the annual meeting.

The annual meeting itself included overview lectures by senior members of the collaboration as well as invited talks by external speakers, addressing emerging and complementary themes in turbulence research. The program encompassed topics such as the discovery and high-precision resolution of unstable singularities using neural-network-based methods, advances in rotating and internal-wave turbulence, and new perspectives on dispersive dynamics. In addition, the workshop highlighted recent progress in dynamical-systems approaches to dispersive equations, including novel applications of KAM theory to the study of periodic and almost-periodic solutions and the construction of invariant tori.

Together, these two events underscored the collaboration’s substantial progress on multiple fronts and reaffirmed its position as a leading contributor to the theory of wave turbulence and its broader scientific applications.

What follows is a description of three major scientific achievements of the collaboration, followed by summaries of the talks presented during the Courant workshop. The collaborative research by Zaher Hani, Yu Deng, and Xiao Ma on the Boltzmann equation represents one of the most significant advancements in kinetic theory in several decades. This work resolves a central, longstanding obstacle that has persisted since Lanford’s foundational 1975 theorem: the rigorous derivation of the Boltzmann equation from Newtonian hard-sphere dynamics on arbitrarily long-time intervals. This monumental achievement resolves a core component of Hilbert’s sixth problem by, for the first time, establishing the long-time validity of the kinetic description beyond the perturbative, short-time regime that had constrained all previous results. The significance of this breakthrough is immense: it provides a mathematically complete and rigorous bridge from the microscopic, reversible dynamics of individual particles to the irreversible, macroscopic laws of statistical physics.

The depth of this contribution lies not merely in the final result but in the entirely novel methodological framework developed to achieve it. Departing from the classical BBGKY hierarchy, the authors introduced a novel diagrammatic, cluster-expansion approach inspired by statistical physics, which strategically reorganizes the hard-sphere dynamics into sums over collision histories encoded by Feynman-type diagrams. To overcome the inevitable divergence of time-series expansions over long intervals, they devised sophisticated layered expansions and a partial re-summation strategy to isolate only the dynamically relevant portions of these histories. Most notably, the team developed a highly nontrivial combinatorial algorithm capable of extracting strong decay estimates from re-collision structures—which had long been recognized as the principal source of error in kinetic limits. This algorithm provides control with gains precisely commensurate to the number of re-collisions, a level of precision that had never been achieved and constitutes the decisive technical innovation enabling the long-time theory.

This work establishes a new and exceptionally high benchmark for rigor, originality, and technical power in mathematical physics. It not only achieves the rigorous resolution of a century-old problem but also establishes a flexible and robust framework that naturally extends to address fluid and wave limits. By opening the door to a unified understanding of kinetic phenomena across diverse particle and wave systems, this contribution fundamentally reshapes the field. In summary, this achievement places Zaher Hani and his collaborators among the very small number of mathematicians who have successfully combined conceptual clarity with extraordinary technical mastery to secure results of lasting foundational importance.

The work of the collaboration PI Tristan Buckmaster and coauthors on the discovery of unstable singularities and resolving their sharp gradients to machine precision via neural networks represents a profound conceptual breakthrough where analysis, computation, and nonlinear partial differential equations (PDEs) converge. This research successfully tackles a longstanding challenge in fluid and dispersive dynamics: the reliable identification and high-precision resolution of unstable singular and coherent structures that are inaccessible to traditional analytical methods and extremely fragile under numerical perturbation. The authors introduce a novel, systematic computational framework that is fundamentally mathematically informed, moving beyond mere exploration. This is achieved by embedding inductive biases—such as scaling laws and self-similar symmetries—drawn from the underlying PDE structure directly into the neural network architecture. This principled methodology leads to critical discoveries, including entirely new families of unstable self-similar solutions in cornerstone systems such as the Euler, Boussinesq, and incompressible porous media equations. These solutions had previously eluded both rigorous analysis and standard numerical detection. Concurrently, the framework demonstrates remarkable versatility by resolving unstable coherent structures, such as solitons and vortical configurations in nonlinear Schrödinger equations. Instability typically undermines conventional numerical schemes in these cases. Crucially, these computations achieve near-machine precision, on the order of 10-8, which elevates the findings from heuristic evidence to a level of rigor suitable for computer-assisted proofs.

In essence, this body of work establishes a new paradigm for the interaction between rigorous analysis and computation in nonlinear PDEs. It proves that carefully designed learning architectures can function not as substitutes for mathematical insight, but as powerful amplifiers capable of uncovering phenomena beyond the reach of current theory while remaining firmly grounded within it. This framework provides the essential, high-precision numerical foundation required to validate unstable singular dynamics and successfully integrate them into the broader program of rigorous mathematical understanding. This sets a new benchmark for computational methods in fundamental fluid mechanics and dispersive analysis.

Recent research by Michal Shavit (collaboration postdoc), Oliver Bühler, and Jalal Shatah has made significant strides in the theoretical understanding of stratified and rotating fluids. Their key insight was recognizing that the weak-rotation limit of the stratified–rotating Boussinesq equations is inherently singular. By identifying the appropriate asymptotic regime for internal-wave dynamics, they overcame a central theoretical obstacle that had long hindered the field. This critical understanding enabled them to address, from first principles, the longstanding problem of explaining the empirically observed, broadband Garrett–Munk (GM) spectrum directly from the governing equations of fluid motion.

Building upon their earlier kinetic theory for weakly interacting two-dimensional internal–inertial gravity waves, the authors developed a unified framework that rigorously incorporates both stratification and rotation. Within this framework, they successfully identified the unique isotropic, scale-invariant wave spectrum that sustains a constant energy flux across scales. When this rigorously derived spectrum is expressed using the observational variables standard in oceanography, its asymptotic scaling recovers the classic GM form in the weak-rotation limit. This decisive connection between a rigorous kinetic description and the empirically observed GM spectrum represents a critical step toward a first principles theory of internal-wave turbulence.

The strength of this work lies in its profound structural coherence and mathematical rigor. The formulation is rooted in the non-canonical Hamiltonian structure of the Boussinesq equations and strategically exploits the weak-rotation limit as a natural regularization mechanism for problematic wave interactions near the slow manifold, where standard perturbation approaches typically break down. This approach results in the first self-consistent kinetic framework for internal-wave turbulence that is both mathematically well-posed and physically compatible with oceanic observations. Complementary work by Germain and Xiang has further solidified this theory by rigorously establishing the well-posedness of the resulting kinetic equation once the appropriate regularization around slow modes is incorporated. The program is further distinguished by its close integration with concurrent experimental efforts, such as the relocation of the stratified water-tank facility to a rotating platform by Cortet, Mordant, and Boury, enabling controlled laboratory testing in the precise weak-rotation regime identified by the theory. This convergence of analysis, rigorous mathematics, and targeted experiment positions the work as a foundational contribution to the modern theory of internal-wave turbulence.

Mean-Field Limits and Modulated Energy Methods

A central theme in this body of work is the rigorous derivation of mean-field limits for systems of particles with singular interactions—notably Coulomb and Riesz types. These systems are governed by gradient flows, conservative flows, and may include stochastic (noisy) effects. The modulated energy method is introduced as a tool to quantify convergence from a discrete particle system to a continuum PDE limit. At the heart of this approach lies a commutator-type functional inequality, which has seen significant recent progress. Global-in-time convergence is also addressed.

Renormalization and Wave Turbulence Theory

Another topic featured at the meeting is the application of renormalization techniques, traditionally used in quantum field theory, to the theory of wave turbulence. This perspective allows for a unified framework across a wide range of nonlinearities—leading to the development of a renormalized kinetic equation. This integro-differential equation can model wave turbulence from weak to strong regimes and supports a new classification of universality classes based on the role of nonlinear interaction enhancement or suppression.

Energy Cascades and Forced Nonlinear Wave Equations

A more elementary but insightful approach is presented to explore energy cascades in forced nonlinear wave equations, including the nonlinear Schrödinger and shallow water equations. By carefully choosing smooth forcing terms, quasi-singular solutions exhibiting power-law decay in Fourier space, whose exponents depend on the system, are constructed. Some stability results are also provided, ensuring that these power-law behaviors persist under perturbations.

Computational and Machine Learning Approaches to PDEs

The meeting will also include results that applies Physics-Informed Neural Networks (PINNs) to discover novel solutions to nonlinear PDEs, including unstable ones that are otherwise difficult to obtain. These computational insights are then translated into rigorous mathematical proofs using computer-assisted methods. While inspired by classical fluid equations (Euler, Navier-Stokes), the methodologies have broad applicability across PDEs.

Boundary Conditions, Energy Spectra & Number Theory

It is common practice among mathematicians and physicists to assume periodic boundary conditions when analyzing the energy spectrum of evolutionary equations. However, an important question arises: how do these boundary conditions affect for example energy transfer in nonlinear Schrödinger equations? Interestingly, this investigation reveals unexpected connections with analytic number theory and leads to the construction of special solutions that display distinct energy transfer behavior.

Understanding Ocean Mixing: Quantifying Turbulence Across Scales

One of the themes presented by experimentalists concerns the ocean mixing processes and how crucial they are in shaping the global distribution of heat, energy, chemical tracers, and marine life, with direct implications for climate dynamics. However, significant uncertainty remains regarding the rates and mechanisms of mixing across a wide range of spatial and temporal scales. Scientists combine traditional approaches with innovative theoretical and observational techniques to quantify turbulence and mixing in the ocean, aiming to uncover the underlying processes that drive these dynamics at both large and small scales.

Nonlinear and Multiscale Fluid Dynamics

There are fundamental problems in fluid dynamics, of relevance to both environmental and industrial applications, that require placing strong emphasis on nonlinear and multiscale systems. These include the statistical behavior of ocean waves, interactions with floating ice sheets, gas exchange driven by surface-breaking waves, spray formation and transport in the atmosphere, and strategies for mitigating oil spills.

To investigate these phenomena, researchers design and conduct complementary laboratory and numerical experiments aimed at developing simplified physical models that capture the key dynamics of each process. Particular attention is given to the study of breaking waves, bubble dynamics in turbulent flows, gas transfer mechanisms, and the generation of spray through bubble bursting.

Tristan Buckmaster
New York University

Fluid Singularities, Unstable PDE Solutions and Computer-Assisted Proofs
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This talk presents recent work on understanding certain solutions of PDE by combining modern mathematics with classical analysis. Machine learning, particularly physics-informed neural networks (PINNs), is being applied to discover new solutions to nonlinear PDEs with high accuracy. A key aspect is the interplay between these methods to uncover the full spectrum of solutions, significantly, unstable solutions that are challenging to find otherwise.

We will also demonstrate how computer-assisted methods can transform these numerical discoveries into rigorous mathematical proofs. While motivated by fluid mechanics equations such as Euler and Navier-Stokes, the methods discussed have broader applicability to other PDEs.
 

Luc Deike
Princeton University

Broadband Ocean Waves Dynamics and Statistics Involving Wind Forcing and Dissipation by Breaking Through a Multilayer Model
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The ocean wave statistics is controlled by the balance between energy input by the wind, weakly non-linear interaction between waves and energy dissipation due to breaking. I will discuss a multilayer numerical framework which generalizes the single-layer Saint-Venant system into a multi-layer, non-hydrostatic formulation of the Navier-Stokes equations (Popinet, J. Comput. Phys. 2020; Wu et al., J. Fluid Mech. 2023, 2025) allowing to resolve broad banded wave fields in presence of wave breaking and under wind forcing. I will discuss the resulting wave spectra, wave breaking statistics and upper ocean turbulence for both steady and growing wave fields over extended periods of times (several hundreds of peak wave periods).
 

Gregory Falkovich
Weizmann Institute of Science

One Equation to Rule Them All, Weak and Strong
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First, I will briefly review the weak turbulence theory from the perspective of theoretical physics. Second, an even shorter synopsis of quantum field theory will be given with the emphasis on renormalization of system parameters in a state with multi-scale fluctuations. Third and final, I shall describe an emerging theory applying the renormalization approach to wave turbulence. In particular, that will let us derive a closed (integro-differential) equation, which is able to describe wide classes of wave turbulence at any level of nonlinearity, from weak to strong. This so-called renormalized kinetic equation is a new interesting mathematical object. At a physical level, we shall see an emerging classification of the universality classes of wave turbulence, based on whether nonlinear interaction is enhanced or suppressed in the strong-turbulence regime.
 

Erwan Faou
National Institute for Research in Digital Science and Technology

Quasi-singularities and power laws solutions for PDEs with forcing
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In this talk I will address the problem of singularity and quasi-singularity formation in partial differential equations with smooth forcing and smooth coefficients. In simple models like the nonlinear Schroedinger equation and the shallow water equation, I will show how a smooth forcing can yield to solutions exhibiting quasi singularities. While the forcing terms considered are smooth and well localized in Fourier, these solutions have Fourier spectra decaying at power laws rate depending on the algebraic structure of the nonlinearity of the equation. I will then study the stability of such solutions and show some numerical examples. These are joint works with R. Carles (CNRS), L. Martaud and G. Beck (INRIA).
 

Nader Masmoudi
New York University

Recent Advances in Nonlinear Inviscid Damping
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Inviscid damping refers to the long-time decay of velocity perturbations in an ideal fluid, even though there’s no viscosity to dissipate the energy. This phenomenon is similar to the Landau damping in plasma physics. We review some old results and give some more recent advances about nonlinear inviscid damping. In particular, we will discuss the extension of the original result to more general shear flows. We will also discuss the optimality of the regularity spaces involved in some results by showing instability constructions. Joint results with J. Bedrossian, Yu Deng, Weiren Zhao.
 

Sylvia Serfaty
Sorbonne Université and Courant Institute

Mean-Field Limits by the Modulated Energy Method
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We will discuss joint works with Matt Rosenzweig, Antonin Chodron de Courcel, Hung Q. Nguyen, and Elias Hess-Childs, in which we study the question of mean-field limits, or deriving effective evolution equations of PDE type for a system of N points in singular interaction, for instance of Coulomb or Riesz, evolving by gradient flow or conservative flow (such as the point vortex system in 2D), with or without noise.

The convergence to the mean-field limit by the modulated energy method relies on a functional inequality of commutator estimate type. We will discuss various ways of proving the commutator estimate, and recent progress that provides sharp and localizable estimates.

We also discuss the question of obtaining global-in-time convergence and its connection with modulated log-Sobolev inequalities.
 

Gigliola Staffilani
Massachusetts Institute of Technology

Some Mathematically Rigorous Results in Wave Turbulence Theory
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In this talk, we give an overview of some old and new mathematical advances in wave turbulence theory. We start with the original perspective of Bourgain on the study of the energy spectrum for a periodic nonlinear Schrödinger equation via growth of Sobolev norms. Then we move to the wave kinetic equations as effective equations for the energy spectrum, and we give examples of rigorous proofs of condensate growth and energy transfer. Some of the work presented is in collaboration with N. Camps and B. M. Tran.
 

Stephanie Waterman
University of British Columbia

Filling in the Map: Understanding Arctic Ocean Mixing Rates, Mechanisms, Space-Time Variability, and Decadal-Scale Trends from Ocean Observations
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The rates and mechanisms of ocean mixing are important controls on how the oceans function; yet, our understanding of mixing in the ocean is significantly limited by complex variability in mixing rates and processes, and by a scarcity of direct observations. In the Arctic Ocean, the challenges of understanding ocean mixing are significant: mixing measurements are especially sparse, and latitude, ice, and stratification make the ocean mixing environment unique.

In this talk, I’ll discuss our group’s work that uses a variety of ocean observations and observational methods to better understand Arctic Ocean mixing rates, mechanisms, space-time variability, and decadal-scale trends. Specifically, I will consolidate results from different studies to provide statistical characterizations of Arctic Ocean mixing rate and mixing flux distributions in time and space over a range of scales, as well as insights into the mechanisms driving and/or modulating the observed variability. I’ll argue that mixing in the Arctic Ocean spans multiple regimes with several important consequences: care is needed when inferring mixing rates using standard methods, the mixing of heat may be significantly enhanced above expectation, and the competition between stabilizing and destabilizing regimes may be an important control on stratification and ultimately mixing regime prevalence itself.

Watch a playlist of all presentations from this meeting here.