Integrable Probability and Universality in Mathematical Physics and Machine Learning

这一建议侧重于研究某些概率模型(随机矩阵、相互作用的粒子系统)和可积系统理论(例如非线性波动方程)之间的临界现象中的普遍行为。可积系统是一类超定偏微分方程组，其解可以通过适当的非线性变换(散射变换)进行彻底的分析，从而将复杂的非线性动力学简化为线性动力学。在某些临界(例如相变)或渐近(例如大时间、连续极限)的区域中，可积系统模型或随机矩阵系综显示出显著的普适性模式，使得它们的行为独立于初始数据(即使在存在随机性的情况下)或它们所基于的概率分布的具体细节。这项工作的动机来自于当前物理兴趣的大量模型，包括晶体和聚合物生长现象、相互避免多个随机游动、光学中的非线性现象、聚合物、超导体(例如玻色-爱因斯坦凝聚体)和流体(最显著的是流浪波)。这项理论研究还将应用于理解机器学习(ML)现象和模型，通过迫切需要的严格框架来描述它们。所有这些实例的主要策略将是根据一个特定的边值问题，即所谓的Riemann-Hilbert问题(RHP)及其与Painlevé方程和等单调tau函数的几何联系来重新表述它们。研究议程分为以下平行的研究方向：(1)分析在某些渐近区域内行列式点过程和随机矩阵模型的统计量的普适性及其几何解释；(2)分析具有随机性的色散可积偏微分方程组(例如Korteweg-de Vries，非线性薛定谔方程)的渐近行为：特别是孤子和孤子气体；(3)研究过参数ML模型的普适性方面：研究在某些渐近或临界区域(例如长时间训练、无限深/宽网络、平均场区域、神经切核区域等)中可积结构的出现。并描述了深度模型的泛化风险曲线。

This proposal focuses on the study of universal behaviors in critical phenomena at the interface between some probabilistic models (random matrices, interacting particle systems) and the theory of integrable systems (e.g. nonlinear wave equations). Integrable systems consist in a class of overdetermined sets of partial differential equations whose solutions can be thoroughly analyzed via a suitable nonlinear transformation (Scattering Transform) that reduces the complicated, nonlinear dynamics into a linear one. In certain critical (e.g. phase transition) or asymptotic (e.g. large-time, continuum limit) regimes, Integrable System models or Random Matrix ensembles display remarkable universality patterns, such that their behaviour become independent on the initial data (even in the presence of randomness) or on the specific details of the probability distribution they are based on. The motivations for this work derive from a plethora of models of current physical interest, including crystal and polymer growth phenomena, mutually-avoiding multiple random-walkers, nonlinear phenomena in optics, polymers, superconductors (e.g. Bose-Einstein condensate) and fluids (most notably, rogue waves). This theoretical research will additionally be applied to understand Machine Learning (ML) phenomena and models, by describing them through a much needed rigorous framework. The main strategy for all these instances will be their reformulation in terms of a particular boundary value problem, the so-called Riemann-Hilbert problem (RHP) and its geometrical connection with Painlevé equations and isomonodromic tau functions. The research agenda is organized into the following parallel directions of research: (1) analyze universality properties of statistical quantities of determinantal point processes and random matrix models within certain asymptotic regimes, and their geometrical interpretation; (2) analyze asymptotic behaviour of dispersive integrable PDEs (e.g. Korteweg-de Vries, Nonlinear Schrödinger equations) with randomness: in particular, solitons and soliton gasses; (3) study universality aspects of overparametrized ML models: investigate the emerging of integrable structures in certain asymptotic or critical regimes (e.g. long time training, infinitely deep/wide networks, mean-field regime, neural tangent kernel regime, etc.) and describe the generalization risk curve of deep models.