Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach

There exist numerous nonlinear partial differential equations in dispersive, as well as in dissipative systems which are of broad applicability in a wide range of sp atio-temporally dependent physical and biological settings. For instance, on the dispersive si de, the Nonlinear Schr odinger (NLS) model [1, 2] has been argued to be of relevance to optical [3, 4 ] and atomic physics [5, 6, 7] to plasma [8, 9] as well as fluid research [10, 9] and even to bio logical applications such as the DNA denaturation [11, 12]. This work has also been central to the seminal contributions of S. Aubry, especially in connection to discrete solitons and br eathers [13, 14, 15]. In a similar vein, in dissipative, reaction-diffusion systems one of the central m odels has been the Fisher-KPP (FKPP) equation which was originally conceived in the context of sp atial spread of advantageous alleles, and has since been employed to model species invasion, popul ation dispersal, and ecological front propagation [16, 17]. The FKPP model has also constituted fo r almost a century now a hallmark of reaction-diffusion models with a wide range of application s to cancer modeling, wound healing, flame propagation and a diverse further host of applications up to this day [18, 19] While such classical PDE models have for decades now been at t he center of initially analytical and subsequently computational studies, over the past deca de or so, a new suite of data-driven tools and techniques has met with explosive growth offering a n ew avenue and a fresh perspective enabling unprecedented developments.