arXiv subjects

Omer San

Publications and source records attributed to Omer San.

At least 55 records · Page 3Linked to original sources

arXiv2022

Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems

A central challenge in the computational modeling and simulation of a multitude of science applications is to achieve robust and accurate closures for their coarse-grained representations due to underlying highly nonlinear multiscale interactions. These closure models are common in many nonlinear spatiotemporal systems to account for losses due to reduced order representations, including many transport phenomena in fluids. Previous data-driven closure modeling efforts have mostly focused on supervised learning approaches using high fidelity simulation data. On the other hand, reinforcement learning (RL) is a powerful yet relatively uncharted method in spatiotemporally extended systems. In this study, we put forth a modular dynamic closure modeling and discovery framework to stabilize the Galerkin projection based reduced order models that may arise in many nonlinear spatiotemporal dynamical systems with quadratic nonlinearity. However, a key element in creating a robust RL agent is to introduce a feasible reward function, which can be constituted of any difference metrics between the RL model and high fidelity simulation data. First, we introduce a multi-modal RL (MMRL) to discover mode-dependant closure policies that utilize the high fidelity data in rewarding our RL agent. We then formulate a variational multiscale RL (VMRL) approach to discover closure models without requiring access to the high fidelity data in designing the reward function. Specifically, our chief innovation is to leverage variational multiscale formalism to quantify the difference between modal interactions in Galerkin systems. Our results in simulating the viscous Burgers equation indicate that the proposed VMRL method leads to robust and accurate closure parameterizations, and it may potentially be used to discover scale-aware closure models for complex dynamical systems.

cs.LG↗

arXiv2022

Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approach

Upcoming technologies like digital twins, autonomous, and artificial intelligent systems involving safety-critical applications require models which are accurate, interpretable, computationally efficient, and generalizable. Unfortunately, the two most commonly used modeling approaches, physics-based modeling (PBM) and data-driven modeling (DDM) fail to satisfy all these requirements. In the current work, we demonstrate how a hybrid approach combining the best of PBM and DDM can result in models which can outperform them both. We do so by combining partial differential equations based on first principles describing partially known physics with a black box DDM, in this case, a deep neural network model compensating for the unknown physics. First, we present a mathematical argument for why this approach should work and then apply the hybrid approach to model two dimensional heat diffusion problem with an unknown source term. The result demonstrates the method's superior performance in terms of accuracy, and generalizability. Additionally, it is shown how the DDM part can be interpreted within the hybrid framework to make the overall approach reliable.

cs.LG↗

arXiv2022

Equation-free surrogate modeling of geophysical flows at the intersection of machine learning and data assimilation

There is a growing interest in developing data-driven reduced-order models for atmospheric and oceanic flows that are trained on data obtained either from high-resolution simulations or satellite observations. The data-driven models are non-intrusive in nature and offer significant computational savings compared to large-scale numerical models. These low-dimensional models can be utilized to reduce the computational burden of generating forecasts and estimating model uncertainty without losing the key information needed for data assimilation to produce accurate state estimates. This paper aims at exploring an equation-free surrogate modeling approach at the intersection of machine learning and data assimilation in Earth system modeling. With this objective, we introduce an end-to-end non-intrusive reduced-order modeling (NIROM) framework equipped with contributions in modal decomposition, time series prediction, optimal sensor placement, and sequential data assimilation. Specifically, we use proper orthogonal decomposition (POD) to identify the dominant structures of the flow, and a long short-term memory network to model the dynamics of the POD modes. The NIROM is integrated within the deterministic ensemble Kalman filter (DEnKF) to incorporate sparse and noisy observations at optimal sensor locations obtained through QR pivoting. The feasibility and the benefit of the proposed framework are demonstrated for the NOAA Optimum Interpolation Sea Surface Temperature (SST) V2 dataset. Our results indicate that the NIROM is stable for long-term forecasting and can model dynamics of SST with a reasonable level of accuracy. Furthermore, the prediction accuracy of the NIROM gets improved by one order of magnitude by the DEnKF algorithm. This work provides a way forward toward transitioning these methods to fuse information from Earth system models and observations to achieve accurate forecasts.

physics.flu-dyn↗

arXiv2022

Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling

We propose a new physics guided machine learning (PGML) paradigm that leverages the variational multiscale (VMS) framework and available data to dramatically increase the accuracy of reduced order models (ROMs) at a modest computational cost. The hierarchical structure of the ROM basis and the VMS framework enable a natural separation of the resolved and unresolved ROM spatial scales. Modern PGML algorithms are used to construct novel models for the interaction among the resolved and unresolved ROM scales. Specifically, the new framework builds ROM operators that are closest to the true interaction terms in the VMS framework. Finally, machine learning is used to reduce the projection error and further increase the ROM accuracy. Our numerical experiments for a two-dimensional vorticity transport problem show that the novel PGML-VMS-ROM paradigm maintains the low computational cost of current ROMs, while significantly increasing the ROM accuracy.

physics.flu-dyn↗

arXiv2022

Physics guided neural networks for modelling of non-linear dynamics

The success of the current wave of artificial intelligence can be partly attributed to deep neural networks, which have proven to be very effective in learning complex patterns from large datasets with minimal human intervention. However, it is difficult to train these models on complex dynamical systems from data alone due to their low data efficiency and sensitivity to hyperparameters and initialisation. This work demonstrates that injection of partially known information at an intermediate layer in a DNN can improve model accuracy, reduce model uncertainty, and yield improved convergence during the training. The value of these physics-guided neural networks has been demonstrated by learning the dynamics of a wide variety of nonlinear dynamical systems represented by five well-known equations in nonlinear systems theory: the Lotka-Volterra, Duffing, Van der Pol, Lorenz, and Henon-Heiles systems.

cs.LG↗

arXiv2022

Reduced Order Model Closures: A Brief Tutorial

In this paper, we present a brief tutorial on reduced order model (ROM) closures. First, we carefully motivate the need for ROM closure modeling in under-resolved simulations. Then, we construct step by step the ROM closure model by extending the classical Galerkin framework to the spaces of resolved and unresolved scales. Finally, we develop the data-driven variational multiscale ROM closure and then we test it in fluid flow simulations. Our tutorial on ROM closures is structured as a sequence of questions and answers, and is aimed at first year graduate students and advanced undergraduate students. Our goal is not to explain the "how," but the "why." That is, we carefully explain the principles used to develop ROM closures, without focusing on particular approaches. Furthermore, we try to keep the technical details to a minimum and describe the general ideas in broad terms while citing appropriate references for details.

math.NA↗

arXiv2022

Sketching Methods for Dynamic Mode Decomposition in Spherical Shallow Water Equations

Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.

math.NA↗

arXiv2022

Frame invariant neural network closures for Kraichnan turbulence

Numerical simulations of geophysical and atmospheric flows have to rely on parameterizations of subgrid scale processes due to their limited spatial resolution. Despite substantial progress in developing parameterization (or closure) models for subgrid scale (SGS) processes using physical insights and mathematical approximations, they remain imperfect and can lead to inaccurate predictions. In recent years, machine learning has been successful in extracting complex patterns from high-resolution spatio-temporal data, leading to improved parameterization models, and ultimately better coarse grid prediction. However, the inability to satisfy known physics and poor generalization hinders the application of these models for real-world problems. In this work, we propose a frame invariant closure approach to improve the accuracy and generalizability of deep learning-based subgrid scale closure models by embedding physical symmetries directly into the structure of the neural network. Specifically, we utilized specialized layers within the convolutional neural network in such a way that desired constraints are theoretically guaranteed without the need for any regularization terms. We demonstrate our framework for a two-dimensional decaying turbulence test case mostly characterized by the forward enstrophy cascade. We show that our frame invariant SGS model (i) accurately predicts the subgrid scale source term, (ii) respects the physical symmetries such as translation, Galilean, and rotation invariance, and (iii) is numerically stable when implemented in coarse-grid simulation with generalization to different initial conditions and Reynolds number. This work builds a bridge between extensive physics-based theories and data-driven modeling paradigms, and thus represents a promising step towards the development of physically consistent data-driven turbulence closure models.

physics.flu-dyn↗

arXiv2022

Hyperparameter Search using Genetic Algorithm for Surrogate Modeling of Geophysical Flows

The computational models for geophysical flows are computationally very expensive to employ in multi-query tasks such as data assimilation, uncertainty quantification, and hence surrogate models sought to alleviate the computational burden associated with these full order models. Researchers have started applying machine learning algorithms, particularly neural networks, to build data-driven surrogate models for geophysical flows. The performance of the neural network highly relies upon its architecture design and selection of other hyperparameters. These neural networks are usually manually designed through trial and error to maximize their performance. This often requires domain knowledge of the neural network as well as the problems of interest. This limitation can be addressed by using an evolutionary algorithm to automatically design architecture and select optimal hyperparameters of the neural network. In this paper, we apply the genetic algorithm to effectively design the long short-term memory (LSTM) neural network to build the non-intrusive reduced order model of the sea-surface temperature field.

physics.comp-ph↗

arXiv2021

Risk-based implementation of COLREGs for autonomous surface vehicles using deep reinforcement learning

Autonomous systems are becoming ubiquitous and gaining momentum within the marine sector. Since the electrification of transport is happening simultaneously, autonomous marine vessels can reduce environmental impact, lower costs, and increase efficiency. Although close monitoring is still required to ensure safety, the ultimate goal is full autonomy. One major milestone is to develop a control system that is versatile enough to handle any weather and encounter that is also robust and reliable. Additionally, the control system must adhere to the International Regulations for Preventing Collisions at Sea (COLREGs) for successful interaction with human sailors. Since the COLREGs were written for the human mind to interpret, they are written in ambiguous prose and therefore not machine-readable or verifiable. Due to these challenges and the wide variety of situations to be tackled, classical model-based approaches prove complicated to implement and computationally heavy. Within machine learning (ML), deep reinforcement learning (DRL) has shown great potential for a wide range of applications. The model-free and self-learning properties of DRL make it a promising candidate for autonomous vessels. In this work, a subset of the COLREGs is incorporated into a DRL-based path following and obstacle avoidance system using collision risk theory. The resulting autonomous agent dynamically interpolates between path following and COLREG-compliant collision avoidance in the training scenario, isolated encounter situations, and AIS-based simulations of real-world scenarios.

cs.RO↗

arXiv2021

Deep neural network enabled corrective source term approach to hybrid analysis and modeling

In this work, we introduce, justify and demonstrate the Corrective Source Term Approach (CoSTA) -- a novel approach to Hybrid Analysis and Modeling (HAM). The objective of HAM is to combine physics-based modeling (PBM) and data-driven modeling (DDM) to create generalizable, trustworthy, accurate, computationally efficient and self-evolving models. CoSTA achieves this objective by augmenting the governing equation of a PBM model with a corrective source term generated using a deep neural network. In a series of numerical experiments on one-dimensional heat diffusion, CoSTA is found to outperform comparable DDM and PBM models in terms of accuracy -- often reducing predictive errors by several orders of magnitude -- while also generalizing better than pure DDM. Due to its flexible but solid theoretical foundation, CoSTA provides a modular framework for leveraging novel developments within both PBM and DDM. Its theoretical foundation also ensures that CoSTA can be used to model any system governed by (deterministic) partial differential equations. Moreover, CoSTA facilitates interpretation of the DNN-generated source term within the context of PBM, which results in improved explainability of the DNN. These factors make CoSTA a potential door-opener for data-driven techniques to enter high-stakes applications previously reserved for pure PBM.

cs.NE↗

arXiv2021

Nonlinear proper orthogonal decomposition for convection-dominated flows

Autoencoder techniques find increasingly common use in reduced order modeling as a means to create a latent space. This reduced order representation offers a modular data-driven modeling approach for nonlinear dynamical systems when integrated with a time series predictive model. In this letter, we put forth a nonlinear proper orthogonal decomposition (POD) framework, which is an end-to-end Galerkin-free model combining autoencoders with long short-term memory networks for dynamics. By eliminating the projection error due to the truncation of Galerkin models, a key enabler of the proposed nonintrusive approach is the kinematic construction of a nonlinear mapping between the full-rank expansion of the POD coefficients and the latent space where the dynamics evolve. We test our framework for model reduction of a convection-dominated system, which is generally challenging for reduced order models. Our approach not only improves the accuracy, but also significantly reduces the computational cost of training and testing.

physics.flu-dyn↗

arXiv2021

Multi-fidelity information fusion with concatenated neural networks

Recently, computational modeling has shifted towards the use of deep learning, and other data-driven modeling frameworks. Although this shift in modeling holds promise in many applications like design optimization and real-time control by lowering the computational burden, training deep learning models needs a huge amount of data. This big data is not always available for scientific problems and leads to poorly generalizable data-driven models. This gap can be furnished by leveraging information from physics-based models. Exploiting prior knowledge about the problem at hand, this study puts forth a concatenated neural network approach to build more tailored, effective, and efficient machine learning models. For our analysis, without losing its generalizability and modularity, we focus on the development of predictive models for laminar and turbulent boundary layer flows. In particular, we combine the self-similarity solution and power-law velocity profile (low-fidelity models) with the noisy data obtained either from experiments or computational fluid dynamics simulations (high-fidelity models) through a concatenated neural network. We illustrate how the knowledge from these simplified models results in reducing uncertainties associated with deep learning models. The proposed framework produces physically consistent models that attempt to achieve better generalization than data-driven models obtained purely based on data. While we demonstrate our framework for a problem relevant to fluid mechanics, its workflow and principles can be adopted for many scientific problems where empirical models are prevalent. In line with grand demands in novel physics-guided machine learning principles, this work builds a bridge between extensive physics-based theories and data-driven modeling paradigms and paves the way for using hybrid modeling approaches for next-generation digital twin technologies.

physics.flu-dyn↗

arXiv2021

On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr-Sommerfeld stability equation and numerous vortex models, of which the von Kármán vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [N. Aubry, P. Holmes, J. Lumley, and E. Stone, Journal of Fluid Mechanics, 192, 115-173 (1988)] pioneered data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data was used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM. But data has made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

physics.flu-dyn↗

arXiv2021

A nonintrusive hybrid neural-physics modeling of incomplete dynamical systems: Lorenz equations

This work presents a hybrid modeling approach to data-driven learning and representation of unknown physical processes and closure parameterizations. These hybrid models are suitable for situations where the mechanistic description of dynamics of some variables is unknown, but reasonably accurate observational data can be obtained for the evolution of the state of the system. In this work, we propose machine learning to account for missing physics and then data assimilation to correct the prediction. In particular, we devise an effective methodology based on a recurrent neural network to model the unknown dynamics. A long short-term memory (LSTM) based correction term is added to the predictive model in order to take into account hidden physics. Since LSTM introduces a black-box approach for the unknown part of the model, we investigate whether the proposed hybrid neural-physical model can be further corrected through a sequential data assimilation step. We apply this framework to the weakly nonlinear Lorenz model that displays quasiperiodic oscillations, the highly nonlinear Lorenz model, and two-scale Lorenz model. The hybrid neural-physics model yields accurate results for the weakly nonlinear Lorenz model with the predicted state close to the true Lorenz model trajectory. For the highly nonlinear Lorenz model and the two-scale Lorenz model, the hybrid neural-physics model deviates from the true state due to the accumulation of prediction error from one time step to the next time step. The ensemble Kalman filter approach takes into account the prediction error and updates the diverged prediction using available observations in order to provide a more accurate state estimate. The successful synergistic integration of neural network and data assimilation for low-dimensional system shows the potential benefits of the proposed hybrid-neural physics model for complex systems.

physics.comp-ph↗

arXiv2021

A Numerical Investigation of the Lengthscale in the Mixing-Length Reduced Order Model of the Turbulent Channel Flow

In this paper, we propose a novel reduced order model (ROM) lengthscale definition that is based on energy distribution arguments. This novel ROM lengthscale is fundamentally different from the current ROM lengthscales, which are generally based on dimensional arguments. As a first step in the assessment of the new, energy based ROM lengthscale, we compare it with a standard, dimensional based ROM lengthscale in the mixing-length ROM (ML-ROM) simulation of the turbulent channel flow at $Re_τ = 395$. The numerical investigation shows that the energy based ROM lengthscale yields a significantly more stable ML-ROM than the dimensional based ROM lengthscale. The new energy based lengthscale definition could allow the development of scale-aware reduced order modeling strategies that are better suited for flow-specific applications.

physics.flu-dyn↗

arXiv2021

Data assimilation empowered neural network parameterizations for subgrid processes in geophysical flows

In the past couple of years, there is a proliferation in the use of machine learning approaches to represent subgrid scale processes in geophysical flows with an aim to improve the forecasting capability and to accelerate numerical simulations of these flows. Despite its success for different types of flow, the online deployment of a data-driven closure model can cause instabilities and biases in modeling the overall effect of subgrid scale processes, which in turn leads to inaccurate prediction. To tackle this issue, we exploit the data assimilation technique to correct the physics-based model coupled with the neural network as a surrogate for unresolved flow dynamics in multiscale systems. In particular, we use a set of neural network architectures to learn the correlation between resolved flow variables and the parameterizations of unresolved flow dynamics and formulate a data assimilation approach to correct the hybrid model during their online deployment. We illustrate our framework in a set of applications of the multiscale Lorenz 96 system for which the parameterization model for unresolved scales is exactly known, and the two-dimensional Kraichnan turbulence system for which the parameterization model for unresolved scales is not known a priori. Our analysis, therefore, comprises a predictive dynamical core empowered by (i) a data-driven closure model for subgrid scale processes, (ii) a data assimilation approach for forecast error correction, and (iii) both data-driven closure and data assimilation procedures. We show significant improvement in the long-term prediction of the underlying chaotic dynamics with our framework compared to using only neural network parameterizations for future prediction.

physics.comp-ph↗

arXiv2021

Model fusion with physics-guided machine learning

The unprecedented amount of data generated from experiments, field observations, and large-scale numerical simulations at a wide range of spatio-temporal scales have enabled the rapid advancement of data-driven and especially deep learning models in the field of fluid mechanics. Although these methods are proven successful for many applications, there is a grand challenge of improving their generalizability. This is particularly essential when data-driven models are employed within outer-loop applications like optimization. In this work, we put forth a physics-guided machine learning (PGML) framework that leverages the interpretable physics-based model with a deep learning model. The PGML framework is capable of enhancing the generalizability of data-driven models and effectively protect against or inform about the inaccurate predictions resulting from extrapolation. We apply the PGML framework as a novel model fusion approach combining the physics-based Galerkin projection model and long-short term memory (LSTM) network for parametric model order reduction of fluid flows. We demonstrate the improved generalizability of the PGML framework against a purely data-driven approach through the injection of physics features into intermediate LSTM layers. Our quantitative analysis shows that the overall model uncertainty can be reduced through the PGML approach especially for test data coming from a distribution different than the training data. Moreover, we demonstrate that our approach can be used as an inverse diagnostic tool providing a confidence score associated with models and observations. The proposed framework also allows for multi-fidelity computing by making use of low-fidelity models in the online deployment of quantified data-driven models.

physics.comp-ph↗