Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learning

The goal of this work is to address two limitations in autoencoder-based models: latent space interpretability and compatibility with unstructured meshes. This is accomplished here with the development of a novel graph neural network (GNN) autoencoding architecture with demonstrations on complex fluid flow applications. To address the first goal of interpretability, the GNN autoencoder achieves reduction in the number nodes in the encoding stage through an adaptive graph reduction procedure. This reduction procedure essentially amounts to flowfield-conditioned node sampling and sensor identification, and produces interpretable latent graph representations tailored to the flowfield reconstruction task in the form of so-called masked fields. These masked fields allow the user to (a) visualize where in physical space a given latent graph is active, and (b) interpret the time-evolution of the latent graph connectivity in accordance with the time-evolution of unsteady flow features (e.g. recirculation zones, shear layers) in the domain. To address the goal of unstructured mesh compatibility, the autoencoding architecture utilizes a series of multi-scale message passing (MMP) layers, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the masked fields. Analysis of latent graphs produced by the autoencoder for various model settings are conducted using using unstructured snapshot data sourced from large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers.

Differentiable physics-enabled closure modeling for Burgers' turbulence

Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with machine learning to develop closure models for Burgers' turbulence. We consider the 1D Burgers system as a prototypical test problem for modeling the unresolved terms in advection-dominated turbulence problems. We train a series of models that incorporate varying degrees of physical assumptions on an a posteriori loss function to test the efficacy of models across a range of system parameters, including viscosity, time, and grid resolution. We find that constraining models with inductive biases in the form of partial differential equations that contain known physics or existing closure approaches produces highly data-efficient, accurate, and generalizable models, outperforming state-of-the-art baselines. Addition of structure in the form of physics information also brings a level of interpretability to the models, potentially offering a stepping stone to the future of closure modeling.

Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems

In data-driven modeling of spatiotemporal phenomena careful consideration often needs to be made in capturing the dynamics of the high wavenumbers. This problem becomes especially challenging when the system of interest exhibits shocks or chaotic dynamics. We present a data-driven modeling method that accurately captures shocks and chaotic dynamics by proposing a novel architecture, stabilized neural ordinary differential equation (ODE). In our proposed architecture, we learn the right-hand-side (RHS) of an ODE by adding the outputs of two NN together where one learns a linear term and the other a nonlinear term. Specifically, we implement this by training a sparse linear convolutional NN to learn the linear term and a dense fully-connected nonlinear NN to learn the nonlinear term. This is in contrast with the standard neural ODE which involves training only a single NN for learning the RHS. We apply this setup to the viscous Burgers equation, which exhibits shocked behavior, and show better short-time tracking and prediction of the energy spectrum at high wavenumbers than a standard neural ODE. We also find that the stabilized neural ODE models are much more robust to noisy initial conditions than the standard neural ODE approach. We also apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation. In this case, stabilized neural ODEs keep long-time trajectories on the attractor, and are highly robust to noisy initial conditions, while standard neural ODEs fail at achieving either of these results. We conclude by demonstrating how stabilizing neural ODEs provide a natural extension for use in reduced-order modeling by projecting the dynamics onto the eigenvectors of the learned linear term.

Multi-fidelity reinforcement learning framework for shape optimization

Deep reinforcement learning (DRL) is a promising outer-loop intelligence paradigm which can deploy problem solving strategies for complex tasks. Consequently, DRL has been utilized for several scientific applications, specifically in cases where classical optimization or control methods are limited. One key limitation of conventional DRL methods is their episode-hungry nature which proves to be a bottleneck for tasks which involve costly evaluations of a numerical forward model. In this article, we address this limitation of DRL by introducing a controlled transfer learning framework that leverages a multi-fidelity simulation setting. Our strategy is deployed for an airfoil shape optimization problem at high Reynolds numbers, where our framework can learn an optimal policy for generating efficient airfoil shapes by gathering knowledge from multi-fidelity environments and reduces computational costs by over 30\%. Furthermore, our formulation promotes policy exploration and generalization to new environments, thereby preventing over-fitting to data from solely one fidelity. Our results demonstrate this framework's applicability to other scientific DRL scenarios where multi-fidelity environments can be used for policy learning.

AutoDEUQ: Automated Deep Ensemble with Uncertainty Quantification

Deep neural networks are powerful predictors for a variety of tasks. However, they do not capture uncertainty directly. Using neural network ensembles to quantify uncertainty is competitive with approaches based on Bayesian neural networks while benefiting from better computational scalability. However, building ensembles of neural networks is a challenging task because, in addition to choosing the right neural architecture or hyperparameters for each member of the ensemble, there is an added cost of training each model. We propose AutoDEUQ, an automated approach for generating an ensemble of deep neural networks. Our approach leverages joint neural architecture and hyperparameter search to generate ensembles. We use the law of total variance to decompose the predictive variance of deep ensembles into aleatoric (data) and epistemic (model) uncertainties. We show that AutoDEUQ outperforms probabilistic backpropagation, Monte Carlo dropout, deep ensemble, distribution-free ensembles, and hyper ensemble methods on a number of regression benchmarks.

Assessments of model-form uncertainty using Gaussian stochastic weight averaging for fluid-flow regression

We use Gaussian stochastic weight averaging (SWAG) to assess the model-form uncertainty associated with neural-network-based function approximation relevant to fluid flows. SWAG approximates a posterior Gaussian distribution of each weight, given training data, and a constant learning rate. Having access to this distribution, it is able to create multiple models with various combinations of sampled weights, which can be used to obtain ensemble predictions. The average of such an ensemble can be regarded as the `mean estimation', whereas its standard deviation can be used to construct `confidence intervals', which enable us to perform uncertainty quantification (UQ) with regard to the training process of neural networks. We utilize representative neural-network-based function approximation tasks for the following cases: (i) a two-dimensional circular-cylinder wake; (ii) the DayMET dataset (maximum daily temperature in North America); (iii) a three-dimensional square-cylinder wake; and (iv) urban flow, to assess the generalizability of the present idea for a wide range of complex datasets. SWAG-based UQ can be applied regardless of the network architecture, and therefore, we demonstrate the applicability of the method for two types of neural networks: (i) global field reconstruction from sparse sensors by combining convolutional neural network (CNN) and multi-layer perceptron (MLP); and (ii) far-field state estimation from sectional data with two-dimensional CNN. We find that SWAG can obtain physically-interpretable confidence-interval estimates from the perspective of model-form uncertainty. This capability supports its use for a wide range of problems in science and engineering.

Data-Driven Wind Turbine Wake Modeling via Probabilistic Machine Learning

Wind farm design primarily depends on the variability of the wind turbine wake flows to the atmospheric wind conditions, and the interaction between wakes. Physics-based models that capture the wake flow-field with high-fidelity are computationally very expensive to perform layout optimization of wind farms, and, thus, data-driven reduced order models can represent an efficient alternative for simulating wind farms. In this work, we use real-world light detection and ranging (LiDAR) measurements of wind-turbine wakes to construct predictive surrogate models using machine learning. Specifically, we first demonstrate the use of deep autoencoders to find a low-dimensional \emph{latent} space that gives a computationally tractable approximation of the wake LiDAR measurements. Then, we learn the mapping between the parameter space and the (latent space) wake flow-fields using a deep neural network. Additionally, we also demonstrate the use of a probabilistic machine learning technique, namely, Gaussian process modeling, to learn the parameter-space-latent-space mapping in addition to the epistemic and aleatoric uncertainty in the data. Finally, to cope with training large datasets, we demonstrate the use of variational Gaussian process models that provide a tractable alternative to the conventional Gaussian process models for large datasets. Furthermore, we introduce the use of active learning to adaptively build and improve a conventional Gaussian process model predictive capability. Overall, we find that our approach provides accurate approximations of the wind-turbine wake flow field that can be queried at an orders-of-magnitude cheaper cost than those generated with high-fidelity physics-based simulations.

Learning the temporal evolution of multivariate densities via normalizing flows

In this work, we propose a method to learn probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker-Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can learn solutions to non-local Fokker-Planck equations, such as those arising in systems driven by both Brownian and L\'evy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.

Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning

Achieving accurate and robust global situational awareness of a complex time-evolving field from a limited number of sensors has been a longstanding challenge. This reconstruction problem is especially difficult when sensors are sparsely positioned in a seemingly random or unorganized manner, which is often encountered in a range of scientific and engineering problems. Moreover, these sensors can be in motion and can become online or offline over time. The key leverage in addressing this scientific issue is the wealth of data accumulated from the sensors. As a solution to this problem, we propose a data-driven spatial field recovery technique founded on a structured grid-based deep-learning approach for arbitrary positioned sensors of any numbers. It should be noted that the na\"ive use of machine learning becomes prohibitively expensive for global field reconstruction and is furthermore not adaptable to an arbitrary number of sensors. In the present work, we consider the use of Voronoi tessellation to obtain a structured-grid representation from sensor locations enabling the computationally tractable use of convolutional neural networks. One of the central features of the present method is its compatibility with deep-learning based super-resolution reconstruction techniques for structured sensor data that are established for image processing. The proposed reconstruction technique is demonstrated for unsteady wake flow, geophysical data, and three-dimensional turbulence. The current framework is able to handle an arbitrary number of moving sensors, and thereby overcomes a major limitation with existing reconstruction methods. The presented technique opens a new pathway towards the practical use of neural networks for real-time global field estimation.

Deploying deep learning in OpenFOAM with TensorFlow

We outline the development of a data science module within OpenFOAM which allows for the in-situ deployment of trained deep learning architectures for general-purpose predictive tasks. This module is constructed with the TensorFlow C API and is integrated into OpenFOAM as an application that may be linked at run time. Notably, our formulation precludes any restrictions related to the type of neural network architecture (i.e., convolutional, fully-connected, etc.). This allows for potential studies of complicated neural architectures for practical CFD problems. In addition, the proposed module outlines a path towards an open-source, unified and transparent framework for computational fluid dynamics and machine learning.