Session J15: Low-Order Modeling and Machine Learning in Fluid Dynamics: Methods II

Stochastic Generation of Lagrangian Turbulent Signals by Conditional Generative Diffusion Models

Lagrangian turbulence lies at the core of numerous applied and fundamental problems related to the physics of dispersion and mixing in engineering, biofluids, the atmosphere, oceans, and astrophysics. Despite exceptional theoretical, numerical, and experimental efforts conducted over the past 30 years, no existing models are capable of faithfully reproducing statistical and topological properties exhibited by particle trajectories in turbulence. We propose a machine-learning approach, based on a state-of-the-art diffusion model, to generate full particle trajectories in three-dimensional turbulence at high Reynolds numbers, thereby bypassing the need for direct numerical simulations or experiments to obtain reliable Lagrangian data. Our model demonstrates the ability to reproduce most statistical benchmarks across time scales, including the fat-tail distribution for velocity increments, the anomalous power law and the increased intermittency around the dissipative scale. We also discuss the applicability of the same method to reconstruct partially sampled trajectories, highlighting the adaptability of this approach to different scenarios; we discuss applications to both the reconstruction of tracer particles from three-dimensional DNS of turbulent flows, as well as the reconstruction of trajectories of oceanic drifters released by the Global Drifter Program (GDP).   

Denoising and super-resolution of Flow Data by Physics-Informed Markov Random Fields

This presentation will discuss the denoising and super-resolution of fluid flows using a Bayesian inverse formulation for recovering a flow field from sparse and noisy observations. We assume the noisy flow data follows a distribution modelled by a Markov random field biased towards satisfying the discrete approximations of the continuity and vorticity equations, corrupted by additive Gaussian white noise. We show that the resulting maximum a posteriori estimation requires solving a sparse nonlinear regularized least-squares problem, introduced as Optimizing a Discrete Loss (ODIL), whose solution is the super-resolved and denoised image. We derive a simple expression involving sparse finite-difference differentiation matrices for the Jacobian of the problem and solve it with the Gauss-Newton method. Contrary to physics-informed neural network based approaches, this method does not require training data. We present results on a variety of vascular flows motivated by denoising and super-resolution of MRI velocimetry. Within this framework, we showcase how the flow field and pressure recovery errors are influenced by the bias toward continuity and vorticity equations.   

Gaussian-process-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility

Following the promising results of a previous work using artificial neural networks (PROM-ANN) (Barnet et al. 2023 JCP) and the realization that one single hidden layer leads to same order-of-magnitude errors, a combination of a projection-based reduced-order model (PROM) and a Gaussian process (GP) is proposed to mitigate the Kolmogorov barrier to reducibility of parametric and/or highly nonlinear, high-dimensional, physics-based models. The main objective of our PROM-GP concept is to reduce the dimensionality of the online approximation of the solution beyond what is achievable using affine and quadratic approximation manifolds, while maintaining accuracy. As well as for its PROM-ANN counterpart, the training of the GP part does not involve data whose dimension scales with that of the high-dimensional model; and the resulting PROM-GP can be efficiently hyperreduced using any well-established hyperreduction method. The added value of using GPs is in the ability to derive mathematical bounds for the errors, not possible in the case of ANNs. All these features make the present concept particularly well-suited for industry-relevant computational problems. Finally, we demonstrate the computational tractability of its offline stage and the superior wall clock time performance of its online stage for a large-scale, parametric, two-dimensional, model problem that is representative of shock-dominated unsteady flow problems.   

Frequency-domain nonlinear model reduction using SPOD modes

Spectral proper orthogonal decomposition (SPOD) has been shown to effectively identify large-scale spatiotemporal coherent structures in fluids systems that play a key role in the dynamics. We present a frequency-domain-based model reduction method that uses SPOD modes to represent the trajectory of the state and solves a system of algebraic equations for the SPOD coefficients given the initial condition and forcing. A significant advantage of this approach is that by leveraging spatiotemporal correlations, trajectories can be represented to orders-of-magnitude more accuracy using some number of SPOD mode coefficients than they can with the same number of POD mode coefficients. In previous work, we developed a model reduction technique to solve quickly and accurately for the SPOD coefficients in linear systems and found that the method achieved substantially lower error than POD-Galerkin and balanced truncation at the same CPU time. In this talk, we extend the method to nonlinear systems. The method selects the most energetic triadic interactions and uses them to compute the effect of the nonlinearity. In many systems, a small number of the triadic interactions account for most of the nonlinearity, so by excluding all but these high-energy interactions, the online time of the method is substantially reduced. In the case of a non-quadratic nonlinearity, the method uses a hyper-reduction technique to handle the nonlinear term. We show that in both cases, we are able to solve the algebraic system that results quickly and that we recover the accuracy afforded by the spatiotemporal trajectory representation.