5 research outputs found
Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization
We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed.
Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized.
Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework.
Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs
Physics-aware registration based auto-encoder for convection dominated PDEs
We design a physics-aware auto-encoder to specifically reduce the
dimensionality of solutions arising from convection-dominated nonlinear
physical systems. Although existing nonlinear manifold learning methods seem to
be compelling tools to reduce the dimensionality of data characterized by a
large Kolmogorov n-width, they typically lack a straightforward mapping from
the latent space to the high-dimensional physical space. Moreover, the realized
latent variables are often hard to interpret. Therefore, many of these methods
are often dismissed in the reduced order modeling of dynamical systems governed
by the partial differential equations (PDEs). Accordingly, we propose an
auto-encoder type nonlinear dimensionality reduction algorithm. The
unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that
registers the output sequence of the PDEs on a non-uniform
parameter/time-varying grid, such that the Kolmogorov n-width of the mapped
data on the learned grid is minimized. We demonstrate the efficacy and
interpretability of our approach to separate convection/advection from
diffusion/scaling on various manufactured and physical systems.Comment: 10 pages, 6 figure
Interpretable structural model error discovery from sparse assimilation increments using spectral bias-reduced neural networks: A quasi-geostrophic turbulence test case
Earth system models suffer from various structural and parametric errors in
their representation of nonlinear, multi-scale processes, leading to
uncertainties in their long-term projections. The effects of many of these
errors (particularly those due to fast physics) can be quantified in short-term
simulations, e.g., as differences between the predicted and observed states
(analysis increments). With the increase in the availability of high-quality
observations and simulations, learning nudging from these increments to correct
model errors has become an active research area. However, most studies focus on
using neural networks, which while powerful, are hard to interpret, are
data-hungry, and poorly generalize out-of-distribution. Here, we show the
capabilities of Model Error Discovery with Interpretability and Data
Assimilation (MEDIDA), a general, data-efficient framework that uses
sparsity-promoting equation-discovery techniques to learn model errors from
analysis increments. Using two-layer quasi-geostrophic turbulence as the test
case, MEDIDA is shown to successfully discover various linear and nonlinear
structural/parametric errors when full observations are available. Discovery
from spatially sparse observations is found to require highly accurate
interpolation schemes. While NNs have shown success as interpolators in recent
studies, here, they are found inadequate due to their inability to accurately
represent small scales, a phenomenon known as spectral bias. We show that a
general remedy, adding a random Fourier feature layer to the NN, resolves this
issue enabling MEDIDA to successfully discover model errors from sparse
observations. These promising results suggest that with further development,
MEDIDA could be scaled up to models of the Earth system and real observations.Comment: 26 pages, 5+1 figure
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Learning Closed-Form Equations for Subgrid-Scale Closures From High-Fidelity Data: Promises and Challenges
There is growing interest in discovering interpretable, closed-form equations for subgrid-scale (SGS) closures/parameterizations of complex processes in Earth systems. Here, we apply a common equation-discovery technique with expansive libraries to learn closures from filtered direct numerical simulations of 2D turbulence and Rayleigh-Bénard convection (RBC). Across common filters (e.g., Gaussian, box), we robustly discover closures of the same form for momentum and heat fluxes. These closures depend on nonlinear combinations of gradients of filtered variables, with constants that are independent of the fluid/flow properties and only depend on filter type/size. We show that these closures are the nonlinear gradient model (NGM), which is derivable analytically using Taylor-series. Indeed, we suggest that with common (physics-free) equation-discovery algorithms, for many common systems/physics, discovered closures are consistent with the leading term of the Taylor-series (except when cutoff filters are used). Like previous studies, we find that large-eddy simulations with NGM closures are unstable, despite significant similarities between the true and NGM-predicted fluxes (correlations >0.95). We identify two shortcomings as reasons for these instabilities: in 2D, NGM produces zero kinetic energy transfer between resolved and subgrid scales, lacking both diffusion and backscattering. In RBC, potential energy backscattering is poorly predicted. Moreover, we show that SGS fluxes diagnosed from data, presumed the “truth” for discovery, depend on filtering procedures and are not unique. Accordingly, to learn accurate, stable closures in future work, we propose several ideas around using physics-informed libraries, loss functions, and metrics. These findings are relevant to closure modeling of any multi-scale system
Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks
Physics-informed neural networks (PINNs) leverage neural-networks to find the
solutions of partial differential equation (PDE)-constrained optimization
problems with initial conditions and boundary conditions as soft constraints.
These soft constraints are often considered to be the sources of the complexity
in the training phase of PINNs. Here, we demonstrate that the challenge of
training (i) persists even when the boundary conditions are strictly enforced,
and (ii) is closely related to the Kolmogorov n-width associated with problems
demonstrating transport, convection, traveling waves, or moving fronts. Given
this realization, we describe the mechanism underlying the training schemes
such as those used in eXtended PINNs (XPINN), curriculum regularization, and
sequence-to-sequence learning. For an important category of PDEs, i.e.,
governed by non-linear convection-diffusion equation, we propose reformulating
PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed
solution. A parallel architecture with two branches is proposed. One branch
solves for the state variables on the characteristics, and the second branch
solves for the low-dimensional characteristics curves. The proposed
architecture conforms to the causality innate to the convection, and leverages
the direction of travel of the information in the domain. Finally, we
demonstrate that the loss landscapes of LPINNs are less sensitive to the
so-called "complexity" of the problems, compared to those in the traditional
PINNs in the Eulerian framework.Comment: 15 pages, 12 figure