Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches

Data-driven approximations of the Koopman operator are promising for predicting the time evolution of systems characterized by complex dynamics. Among these methods, the approach known as extended dynamic mode decomposition with dictionary learning (EDMD-DL) has garnered significant attention. Here we present a modification of EDMD-DL that concurrently determines both the dictionary of observables and the corresponding approximation of the Koopman operator. This innovation leverages automatic differentiation to facilitate gradient descent computations through the pseudoinverse. We also address the performance of several alternative methodologies. We assess a 'pure' Koopman approach, which involves the direct time-integration of a linear, high-dimensional system governing the dynamics within the space of observables. Additionally, we explore a modified approach where the system alternates between spaces of states and observables at each time step -- this approach no longer satisfies the linearity of the true Koopman operator representation. For further comparisons, we also apply a state space approach (neural ODEs). We consider systems encompassing two and three-dimensional ordinary differential equation systems featuring steady, oscillatory, and chaotic attractors, as well as partial differential equations exhibiting increasingly complex and intricate behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the state space approach offers superior performance compared to the 'pure' Koopman approach where the entire time evolution occurs in the space of observables. When the temporal evolution of the Koopman approach alternates between states and observables at each time step, however, its predictions become comparable to those of the state space approach

Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds

The accurate simulation of complex dynamics in fluid flows demands a substantial number of degrees of freedom, i.e. a high-dimensional state space. Nevertheless, the swift attenuation of small-scale perturbations due to viscous diffusion permits in principle the representation of these flows using a significantly reduced dimensionality. Over time, the dynamics of such flows evolve towards a finite-dimensional invariant manifold. Using only data from direct numerical simulations, in the present work we identify the manifold and determine evolution equations for the dynamics on it. We use an advanced autoencoder framework to automatically estimate the intrinsic dimension of the manifold and provide an orthogonal coordinate system. Then, we learn the dynamics by determining an equation on the manifold by using both a function space approach (approximating the Koopman operator) and a state space approach (approximating the vector field on the manifold). We apply this method to exact coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully resolved simulations for these cases require O(103) and O(105) degrees of freedom respectively, and we build models with two or three degrees of freedom that faithfully capture the dynamics of these flows. For these examples, both the state space and function space time evaluations provide highly accurate predictions of the long-time dynamics in manifold coordinates.Comment: 10 page

Role of kidney stones in renal pelvis flow

International audienceWe examine the time-dependent flow dynamics inside an idealised renal pelvis in the context of a surgical procedure for kidney stone removal, extending previous work by Williams et al. (2020a, 2021), who showed how vortical flow structures can hinder mass transport in a canonical two-dimensional domain. Here, we examine the time-dependent evolution of these vortical flow structures in three-dimensions, and incorporate the presence of rigid kidney stones. We perform direct numerical simulations, solving the transient Navier-Stokes equations in a spherical domain. Our numerical predictions for the flow dynamics in the absence of stones are validated with experimental and 2D numerical data from Williams et al. (2020a), and the governing parameters and flow regimes are chosen carefully in order to satisfy several clinical constraints. The results shed light on the crucial role of flow circulation in the renal cavity and its effect on the trajectories of rigid stones. We demonstrate that stones can either be washed out of the cavity along with the fluid, or be trapped in the cavity via their interaction with vortical flow structures. Additionally, we study the effect of multiple stones in the flow field within the cavity in terms of the kinetic energy, entrapped fluid volume, and the clearance rate of a passive tracer modelled via an advection-diffusion equation. We demonstrate that the flow in the presence of stones features a higher vorticity production within the cavity compared with the stone-free cases. Impact Statement Innovative numerical algorithms for complex fluid flows together with high-performance computing architectures can deliver an in-depth understanding of the flow physics of previously inaccessible problems. In this research we have performed numerical solutions of ureteroscopy flows to provide a better understanding of the dynamics of kidney stones during ureteroscopy, a surgical procedure designed to remove kidney stones. For the first time (to the best of our knowledge), we have illustrated the role of rigid stones inside an idealised renal pelvis. This research has the potential to provide a better understanding of the flow dynamics within the cavity during ureteroscopy and subsequently could lead to better surgical practices