Publications

Trask, Nathaniel A.

Abstract not provided.

Computer Methods in Applied Mechanics and Engineering

You, Huaiqian; Yu, Yue; Trask, Nathaniel A.; Gulian, Mamikon G.; D'Elia, Marta D.

A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity — it is particularly challenging to extract surrogates which are provably well-posed and numerically stable. Our scheme not only yields a convex optimization problem, but also allows extraction of nonlocal models whose kernels may be partially negative while maintaining well-posedness even in small-data regimes. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints to meet the requisite conditions of the nonlocal theory. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels.

Cyr, Eric C.; Gulian, Mamikon G.; Patel, Ravi G.; Perego, Mauro P.; Trask, Nathaniel A.

Abstract not provided.

SIAM Journal on Numerical Analysis

Leng, Yu; Tian, Xiaochuan; Trask, Nathaniel A.; Foster, John T.

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The numerical scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for a linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme, which is stable. In addition, assembling the stiffness matrix of the nonlocal problem requires costly computational resources because high-order Gaussian quadrature is necessary to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares.

CEUR Workshop Proceedings

Patel, Ravi G.; Trask, Nathaniel A.; Gulian, Mamikon G.; Cyr, Eric C.

Second-order optimizers hold intriguing potential for deep learning, but suffer from increased cost and sensitivity to the non-convexity of the loss surface as compared to gradient-based approaches. We introduce a coordinate descent method to train deep neural networks for classification tasks that exploits global convexity of the cross-entropy loss in the weights of the linear layer. Our hybrid Newton/Gradient Descent (NGD) method is consistent with the interpretation of hidden layers as providing an adaptive basis and the linear layer as providing an optimal fit of the basis to data. By alternating between a second-order method to find globally optimal parameters for the linear layer and gradient descent to train the hidden layers, we ensure an optimal fit of the adaptive basis to data throughout training. The size of the Hessian in the second-order step scales only with the number weights in the linear layer and not the depth and width of the hidden layers; furthermore, the approach is applicable to arbitrary hidden layer architecture. Previous work applying this adaptive basis perspective to regression problems demonstrated significant improvements in accuracy at reduced training cost, and this work can be viewed as an extension of this approach to classification problems. We first prove that the resulting Hessian matrix is symmetric semi-definite, and that the Newton step realizes a global minimizer. By studying classification of manufactured two-dimensional point cloud data, we demonstrate both an improvement in validation error and a striking qualitative difference in the basis functions encoded in the hidden layer when trained using NGD. Application to image classification benchmarks for both dense and convolutional architectures reveals improved training accuracy, suggesting gains of second-order methods over gradient descent. A Tensorflow implementation of the algorithm is available at github.com/rgp62/.

CEUR Workshop Proceedings

Huang, Andy H.; Trask, Nathaniel A.; Brissette, Christopher; Hu, Xiaozhe

The data-driven discrete exterior calculus (DDEC) structure provides a novel machine learning architecture for discovering structure-preserving models which govern data, allowing for example machine learning of reduced order models for complex continuum scale physical systems. In this work, we present a Greedy Fiedler Spectral (GFS) partitioning method to obtain a chain complex structure to support DDEC models, incorporating synthetic data obtained from high-fidelity solutions to partial differential equations. We provide justification for the effectiveness of the resulting chain complex and demonstrate its DDEC model trained for Darcy flow on a heterogeneous domain.

Kookjin, Lee K.; Trask, Nathaniel A.; Patel, Ravi G.; Gulian, Mamikon G.; Cyr, Eric C.

Abstract not provided.

Computer Methods in Applied Mechanics and Engineering

Patel, Ravi G.; Trask, Nathaniel A.; Wood, Mitchell A.; Cyr, Eric C.

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.

Patel, Ravi G.; Trask, Nathaniel A.; Wood, Mitchell A.; Cyr, Eric C.

Abstract not provided.

Gross, Ben G.; Kuberry, Paul A.; Trask, Nathaniel A.; Atzberger, Paul J.

Abstract not provided.

Trask, Nathaniel A.

Abstract not provided.

Trask, Nathaniel A.

Abstract not provided.

Computer Methods in Applied Mechanics and Engineering

Leng, Yu; Tian, Xiaochuan; Trask, Nathaniel A.; Foster, John T.

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.

International Conference on Simulation of Semiconductor Processes and Devices, SISPAD

Gao, Xujiao G.; Huang, Andy H.; Trask, Nathaniel A.; Reza, Shahed R.

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.

Kuberry, Paul A.; Trask, Nathaniel A.; Koester, Jacob K.; Bochev, Pavel B.

Abstract not provided.

Gao, Xujiao G.; Huang, Andy H.; Trask, Nathaniel A.; Reza, Shahed R.

Abstract not provided.

ESAIM: Mathematical Modelling and Numerical Analysis

You, Huaiqian; Lu, Xin Y.; Trask, Nathaniel A.; Yu, Yue; Yu, Yue

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

Trask, Nathaniel A.

Abstract not provided.

Gulian, Mamikon G.; Cyr, Eric C.; Patel, Ravi G.; Perego, Mauro P.; Trask, Nathaniel A.

Abstract not provided.

Cyr, Eric C.; Gulian, Mamikon G.; Patel, Ravi G.; Perego, Mauro P.; Trask, Nathaniel A.

Abstract not provided.

Arxiv

Patel, Ravi G.; Trask, Nathaniel A.; Gulian, Mamikon G.; Cyr, Eric C.

Abstract not provided.

Trask, Nathaniel A.

Abstract not provided.

Journal of Computational Physics

Trask, Nathaniel A.; Bochev, Pavel B.; Perego, Mauro P.

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.

Arxiv

Behzadinasab, Masoud B.; Bazilevs, Yuri B.; Trask, Nathaniel A.

Abstract not provided.

Computational Particle Mechanics

Trask, Nathaniel A.; Kuberry, Paul A.

Meshfree discretization of surface partial differential equations is appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in Rd. Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. We provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.