MS03.6: Computational Methods

A significant portion of atmospheric CO2 emissions are absorbed by the ocean, resulting in a decreased pH that is harmful to marine life and ecosystems — a process known as ocean acidification (OA). However, monitoring and forecasting indicators of OA is difficult due to a lack of in-situ measurements, the nonlinearity of the dynamics, and the high costs of computational numerical models. We develop a data-driven framework to model properties that drive OA in the Massachusetts Bay and Stellwagen Bank. In the first step of the framework, we train a neural network on data from a historical physics-based numerical simulation to predict 3D temperature and salinity (x,y,z) from quantities at the surface (x,y). The relationship between 2D surface properties and 3D properties is captured through the in-depth modes obtained from principal component analysis. We use this trained model to estimate real-time 4D temperature and salinity from satellite and buoy surface measurements. Then, we use standard Bayesian regression methods to estimate region-specific relationships for total alkalinity (TA) and pH as a function of temperature and salinity. The model's performance is evaluated using withheld measurements at multiple depths. Furthermore, each step of the framework includes uncertainty quantification which can be used to plan future operations and optimally place measurement stations.

Understanding how, when, and why extreme events occur is an important problem in many fields (e.g. extreme climate events, rogue ocean waves, materials science, etc.). However, because these events are infrequent and/or severe by nature, it is often necessary to collect a large amount of data to understand these events and their relationship to the system. As a result, data sets created from experiments and simulations often contain many nearly repetitive, and thus unnecessary, points. Data sets used for such analysis become even larger for problems in high dimensions. Furthermore, training predictive models with these large data sets requires vast computing time and power. The question becomes how to select a subset of points to reduce training time while accurately representing the distribution of the original data. We present an active learning framework that uses a likelihood-weighted sampling criterion to sequentially select optimal training input points that give rise to outputs in the tails of the distribution (i.e. most relevant to the dynamics of extreme events). To compute the criterion, we use neural network architectures capable of making probabilistic predictions. We test the method by predicting the maximum future wave height of a 1D dispersive nonlinear wave model from a high-dimensional set of initial conditions. The likelihood-weighted search algorithm is able to accurately reproduce the probability density function of the original data sets using a fraction of the original points.

Invariant solutions to the Navier-Stokes equations, known as Exact Coherent Structures (ECSs), are known to represent important physical mechanisms of turbulent flows. Recently a variational optimisation approach based on gradient descent was proposed as a robust numerical method for finding these solutions. However, its practical implementation has proven challenging due to the high dimensionality of the problem and the slow convergence rate. Moreover, the method is not applicable to domains containing walls. In this work the variational optimisation methodology is extended to find time periodic solutions in domains that contain walls using a Galerkin projection approach. The basis used to construct the subspace onto which the projection is performed is derived using Resolvent analysis, which has been shown to provide an efficient representation of the invariant solutions for some wall-bounded flows. By truncating the modes used a significant dimensionality reduction is achieved while retaining much of the important dynamical information. This effectively acts as a preconditioner and thereby increases the convergence rate of the optimiser. Additionally, the effect on the convergence rate when replacing the gradient descent algorithm with quasi-Newton algorithms for the optimisation is analysed. The Rotating Plane Couette Flow (RPCF) is chosen to demonstrate the modified optimisation methodology.

We propose a new computational model for migraine-related cortical spreading depression (CSD) by using a neural field framework. The model is extended from the Wilson-Cowan-Amari formalism [1, 2]. It is based on an excitatory-inhibitory neuronal population pair which is coupled to a potassium concentration variable. This model is spatially extended to a cortical layer patch. Therefore, it can model both the ignition and propagation of CSD.

Quasi-periodic solutions exist widely in nonlinear dynamical systems. Multi-harmonic balance method (MHBM) and variable-coefficient harmonic balance method (VCHBM) are two popular methods for determining the quasi-periodic solutions semi-analytically, where MHBM uses a multi-dimensional Fourier series and VCHBM a variable-coefficient Fourier series. In this work, the intrinsic relationship between the two methods is unveiled in the case for the quasi-periodic solutions consisting of two frequency components. The transform matrices between the algebraic equations of the Fourier coefficients of the two methods are explicitly derived. Particularly, the transform matrices are also applied to the transform of the derivative of Fourier coefficients for nonlinear terms between them. An example is given to demonstrate validity of the transformation between the two methods.

Missions aiming for periodic orbits around the Moon are becoming increasingly attractive, largely due to the influence of NASA's Artemis program. However, chaotic dynamics around these orbits still represents a hindrance to optimal design of such space missions. Therefore, informed mission design needs a deeper understanding of cislunar dynamics. To this end, we propose high-order techniques that make use of differential algebra (DA), which we apply to both well-known and novel problems in astrodynamics. In particular, we compute periodic orbits (POs) families of the circular restricted three-body problem with pseudo-arclength continuation and numerically construct transfer maps from a Poincaré section back to itself. The latter can be used for the analysis of the cislunar chaotic dynamics, because they help explain the behavior of ballistic capture trajectories and their connection with POs.