Publications

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The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. Here, we demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows. For the flow physics we demonstrate this approach leads to a learned operator that generalizes to system characteristics not included in the training sets, such as variable particle sizes, densities, and concentration.

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Component coupling is a crucial part of climate models, such as DOE's E3SM (Caldwell et al., 2019). A common coupling strategy in climate models is for their components to exchange flux data from the previous time-step. This approach effectively performs a single step of an iterative solution method for the monolithic coupled system, which may lead to instabilities and loss of accuracy. In this paper we formulate an Interface-Flux-Recovery (IFR) coupling method which improves upon the conventional coupling techniques in climate models. IFR starts from a monolithic formulation of the coupled discrete problem and then uses a Schur complement to obtain an accurate approximation of the flux across the interface between the model components. This decouples the individual components and allows one to solve them independently by using schemes that are optimized for each component. To demonstrate the feasibility of the method, we apply IFR to a simplified ocean–atmosphere model for heat-exchange coupled through the so-called bulk condition, common in ocean–atmosphere systems. We then solve this model on matching and non-matching grids to estimate numerically the convergence rates of the IFR coupling scheme.

In this work, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ≥μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

We present a Physics-Informed Graph Neural Network (pigNN) methodology for rapid and automated compact model development. It brings together the inherent strengths of data-driven machine learning, high-fidelity physics in TCAD simulations, and knowledge contained in existing compact models. In this work, we focus on developing a neural network (NN) based compact model for a non-ideal PN diode that represents one nonlinear edge in a pigNN graph. This model accurately captures the smooth transition between the exponential and quasi-linear response regions. By learning voltage dependent non-ideality factor using NN and employing an inverse response function in the NN loss function, the model also accurately captures the voltage dependent recombination effect. This NN compact model serves as basis model for a PN diode that can be a single device or represent an isolated diode in a complex device determined by topological data analysis (TDA) methods. The pigNN methodology is also applicable to derive reduced order models in other engineering areas.

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This report summarizes the work performed under a one-year LDRD project aiming to enable accurate and robust numerical simulation of partial differential equations for meshes that are of poor quality. Traditional finite element methods use the mesh to both discretize the geometric domain and to define the finite element shape functions. The latter creates a dependence between the quality of the mesh and the properties of the finite element basis that may adversely affect the accuracy of the discretized problem. In this project, we propose a new approach for defining finite element shape functions that breaks this dependence and separates mesh quality from the discretization quality. At the core of the approach is a meshless definition of the shape functions, which limits the purpose of the mesh to representing the geometric domain and integrating the basis functions without having any role in their approximation quality. The resulting non-conforming space can be utilized within a standard discontinuous Galerkin framework providing a rigorous foundation for solving partial differential equations on low-quality meshes. We present a collection of numerical experiments demonstrating our approach in a wide range of settings: strongly coercive elliptic problems, linear elasticity in the compressible regime, and the stationary Stokes problem. We demonstrate convergence for all problems and stability for element pairs for problems which usually require inf-sup compatibility for conforming methods, also referring to a minor modification possible through the symmetric interior penalty Galerkin framework for stabilizing element pairs that would otherwise be traditionally unstable. Mesh robustness is particularly critical for elasticity, and we provide an example that our approach provides a greater than 5x improvement in accuracy and allows for taking an 8x larger stable timestep for a highly deformed mesh, compared to the continuous Galerkin finite element method. The report concludes with a brief summary of ongoing projects and collaborations that utilize or extend the products of this work.

One challenge of using compartmental SEIR models for public health planning is the difficulty in manually tuning parameters to capture behavior reflected in the real-world data. This team conducted initial, exploratory analysis of a novel technique to use physics-informed machine learning tools to rapidly develop data-driven models for physical systems. This machine learning approach may be used to perform data assimilation of compartment models which account for unknown interactions between geospatial domains (i.e. diffusion processes coupling across neighborhoods/counties/states/etc.). Results presented here are early, proof-of-concept ideas that demonstrate initial success in using a physically informed neural network (PINN) model to assimilate data in a compartmental epidemiology model. The results demonstrate initial success and warrant further research and development.

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Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

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A particle-mesh strategy is presented for scalar transport problems which provides diffusion-free advection, conserves mass locally (i.e. cellwise) and exhibits optimal convergence on arbitrary polyhedral meshes. This is achieved by expressing the convective field naturally located on the Lagrangian particles as a mesh quantity by formulating a dedicated particle-mesh projection based via a PDE-constrained optimization problem. Optimal convergence and local conservation are demonstrated for a benchmark test, and the application of the scheme to mass conservative density tracking is illustrated for the Rayleigh–Taylor instability.

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Data fields sampled on irregularly spaced points arise in many science and engineering applications. For regular grids, Convolutional Neural Networks (CNNs) gain benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds using Generalized Moving Least Squares (GMLS). GMLS is a nonparametric meshfree technique for estimating linear bounded functionals from scattered data, and has emerged as an effective technique for solving partial differential equations (PDEs). By parameterizing the GMLS estimator, we obtain learning methods for linear and non-linear operators with unstructured stencils. The requisite calculations are local, embarrassingly parallelizable, and supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform operator regression, develop predictive dynamical models, and obtain feature extractors for engineering quantities of interest. The results show the promise of these architectures as foundations for data-driven model development in scientific machine learning applications.

Motivated by the gap between theoretical optimal approximation rates of deep neural networks (DNNs) and the accuracy realized in practice, we seek to improve the training of DNNs. The adoption of an adaptive basis viewpoint of DNNs leads to novel initializations and a hybrid least squares/gradient descent optimizer. We provide analysis of these techniques and illustrate via numerical examples dramatic increases in accuracy and convergence rate for benchmarks characterizing scientific applications where DNNs are currently used, including regression problems and physics-informed neural networks for the solution of partial differential equations.

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