Rigorously Verified Numerics for High Dimensional Dynamics

非线性动力学在自然界中普遍存在。它在生物学中无处不在，从神经元的电生理特性，到收缩心肌的螺旋波，再到基因调控网络。它在物理学中无处不在，从流体流动中的漩涡运动，到材料中复杂图案的创造，再到天体的和谐运动。它在化学中无处不在，从丰富的反应动力学现象，到动物图案起源时形态发生的化学基础，再到活细胞中复杂的生物化学。在数学上，这些美丽的现象被描述为非线性动力系统的形式常微分方程（ODE），偏微分方程（PDE）和延迟微分方程（DDE）。虽然非线性的存在使分析复杂化，但对于PDE和DDE来说，挑战甚至更大，因为它们自然定义在无限维函数空间上。由于高性能计算机的可用性和复杂软件的快速发展，数值模拟已成为科学家研究这些模型的主要工具。然而，现代科学中计算的普遍性自然导致了输出有效性的基本问题。即使对于常微分方程，如果所研究的系统是混沌的，这个问题也会出现，因为初始条件的微小差异（例如数值模拟中的舍入误差）会产生广泛的差异结果。为了解决这个问题，最近的严格数值领域出现在纯数学和应用数学的交叉点。严格的数值从科学计算，非线性分析，数值分析，应用拓扑学，泛函分析和近似理论中汲取灵感。在这项研究计划中，我们建议在严格的数值领域引入新颖而令人兴奋的技术，以研究有限维和无限维动力系统中的前沿问题，重点是流体，图案形成和天体力学中出现的问题。更确切地说，我们有兴趣回答以下每个问题。我们能严格控制计算偏微分方程柯西问题解时的误差吗？我们能否开发出连接无限维轨道的严格计算，这对我们理解图案形成至关重要？我们能用数学方法证明无穷维连续动力系统中混沌的存在吗？我们能否证明一个长期存在的关于天体力学中舞蹈的延续的猜想？从长远来看，我们是否可以证明纳维尔-斯托克斯方程在某一类初始数据中不会随着时间的推移而发展出奇点，或者我们是否可以确定（通过数值方法--也许通过严格的验证）一个初始数据？

Nonlinear dynamics is ubiquitous in the natural world. It is pervasive in biology, from the electrophysiological properties of neurons, via the spiralling waves in contracting heart muscles, to gene regulatory networks. It is pervasive in physics, from the swirling motions in fluid flows, via the creation of complex patterns in materials, to the harmonious motions of celestial bodies. It is pervasive in chemistry, from the rich reaction kinetics phenomena, via the chemical basis of morphogenesis at the origin of patterns on animals, to the complicated biochemistry in the living cell. Mathematically, these beautiful phenomena are described by nonlinear dynamical systems in the form of ordinary differential equations (ODEs), partial differential equations (PDEs) and delay differential equations (DDEs). While the presence of nonlinearities complicates the analysis, the challenges are even greater for PDEs and DDEs, which are naturally defined on infinite dimensional function spaces. Thanks to the availability of high-performance computers and the rapid development of sophisticated software, numerical simulations have become the primary tool used by scientists to study these models. However, the prevalence of computations in modern day science naturally leads to the fundamental question of validity of the outputs. Even for ODEs, this question arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical simulations) yield widely diverging outcomes. To address this issue, the recent field of rigorous numerics emerged at the intersection of pure and applied mathematics. Rigorous numerics draws inspiration from scientific computing, nonlinear analysis, numerical analysis, applied topology, functional analysis and approximation theory. In this research proposal, we propose to introduce novel and exciting techniques within the field of rigorous numerics to study cutting-edge problems in finite and infinite dimensional dynamical systems, with an emphasis on problems arising in fluids, pattern formation and celestial mechanics. More precisely, we are interested in answering each of the following questions. Can we rigorously control the errors made when computing solutions of Cauchy problems of PDEs? Can we develop rigorous computations for connecting orbits in infinite dimensions, which are crucial for our understanding of pattern formation? Can we mathematically demonstrate the existence of chaos in infinite dimensional continuous dynamical systems? Can we prove a long standing conjecture concerning the continuation of choreographies in celestial mechanics? In the long term, can we show that Navier-Stokes do not develop singularities as time evolves for a certain class of initial data, or can we identify (numerically-perhaps with rigorous validation) an initial data for which it does?