Applied Math Seminar

Synopsis:

Machine Learning (ML) and Artificial Intelligence (AI) are increasingly transforming fusion research, complementing and, in many cases, surpassing traditional statistical tools. These methods are accelerating progress by enabling more accurate modeling, optimized strategies, and enhanced experimental realization [1].

In this seminar, we will focus on the development of interpretable ML-driven metrics for two critical challenges in magnetic confinement fusion: (1) real-time monitoring of proximity to plasma stability boundaries [2,3,4] and (2) the optimization of plasma trajectories [5,6]. By emphasizing interpretability, these approaches not only deliver predictive power but also provide insights that are actionable for control and disruption prevention.

A key element of this work is the use and development of JAX frameworks [7,8], which enable seamless integration of physics equations with neural networks. This hybrid modeling approach accelerates system identification for plasma dynamics, advancing the reliability, efficiency, and scalability of ML-enabled solutions for both existing and next-generation fusion devices – a key focus of the MIT PSFC Disruptions Team (https://disruptions.mit.edu/)

References:

1. Rea J.l of Fusion Energy 44, 39 (2025) https://doi.org/10.1007/s10894-025-00509-z
2. Rea, IAEA Fusion Energy Conference Proceedings EX/P1–25 (2021)
3. Barr et al., Nucl. Fusion 61, 126019 (2021)
4. Maris, Rea et al., Nucl. Fusion 65, 016051 (2025)
5. Wang, Rea et al., Comm. Physics (2025) https://www.nature.com/articles/s42005-025-02146-6
6. Wang, Pau, Rea et al., Nature Communications (accepted, 2025) https://arxiv.org/pdf/2502.12327v2
7. Bradbury et al., (2018) http://github.com/jax-ml/jax
8. Wang et al., IEEE TPS (submitted, 2025)

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Synopsis:

Wannier functions provide localized real-space representations of electronic and photonic states in periodic media, forming the mathematical bridge between band theory and lattice models. Yet constructing the most localized Wannier functions has long relied on heuristic gauge choices and high-dimensional optimization. In this talk, I will describe an analytic framework that corrects Kato’s classical perturbation theory on tori by incorporating geometric quantities such as the Berry connection and curvature, thereby turning a local analytic theory into a global one. This yields a constructive procedure for efficiently computing optimally localized Wannier functions and reveals a unified link among localization, geometry, and topological obstruction. The framework will be illustrated with several numerical examples.

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Synopsis:

The interaction among vortices is a key process in fluid motion. The n-vortex problem, which models the movement of a finite number of vortices in a two-dimensional inviscid fluid, has been studied since the late 1800s and remains relevant due to its strong link to quantum fluid dynamics. A foundational document in this area is Walter Gröbli's 1877 doctoral dissertation. We apply modern tools from dynamical systems and Hamiltonian mechanics to several problems arising from this work. First, we study the linear stability and nonlinear dynamics of the so-called leapfrogging orbit of four vortices, utilizing Hamiltonian reductions and a numerical visualization method known as Lagrangian descriptors. Second, we analyze the scattering of vortex dipoles using tools from geometric mechanics. 

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TBD

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Synopsis:

When wave propagation problems are posed on unbounded domains, most numerical solvers rely on truncation, such as absorbing boundary conditions or perfectly matched layers (PML). I will describe a geometric alternative that solves the original exterior problem numerically: spatial compactification combined with a time shift that places infinity as a characteristic (null) boundary on the computational grid. This method links ideas between Lorentzian geometry and hyperbolic PDEs.

In the frequency domain, the time shift acts as a rephasing that keeps oscillations bounded so the effective wavenumber remains finite after compactification. In the time domain, the transformed first-order system is symmetric hyperbolic with maximally dissipative boundaries, ensuring stability and tracking energy decay.

For practical applications, a Null Infinity Layer (NIL) can wrap any interior mesh by a layer of finite thickness representing the unbounded exterior domain. I will present numerical experiments of benchmark scattering problems showing that NIL has comparable accuracy to PML while providing acces to the far-field.

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