5 research outputs found
Data-driven learning of robust nonlocal physics from high-fidelity synthetic data
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 provably robust data-driven surrogates is a major challenge
for machine learning (ML) approaches, due to nonlinearities and lack of
convexity. Our scheme allows extraction of provably invertible nonlocal models
whose kernels may be partially negative. 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 and ensure
that models are robust, even in small-data regimes. 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.Comment: 32 pages, 10 figures, 3 table
Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws
We show that machine learning can improve the accuracy of simulations of
stress waves in one-dimensional composite materials. We propose a data-driven
technique to learn nonlocal constitutive laws for stress wave propagation
models. The method is an optimization-based technique in which the nonlocal
kernel function is approximated via Bernstein polynomials. The kernel,
including both its functional form and parameters, is derived so that when used
in a nonlocal solver, it generates solutions that closely match high-fidelity
data. The optimal kernel therefore acts as a homogenized nonlocal continuum
model that accurately reproduces wave motion in a smaller-scale, more detailed
model that can include multiple materials. We apply this technique to wave
propagation within a heterogeneous bar with a periodic microstructure. Several
one-dimensional numerical tests illustrate the accuracy of our algorithm. The
optimal kernel is demonstrated to reproduce high-fidelity data for a composite
material in applications that are substantially different from the problems
used as training data
An optimization-based approach to parameter learning for fractional type nonlocal models
Nonlocal operators of fractional type are a popular modeling choice for
applications that do not adhere to classical diffusive behavior; however, one
major challenge in nonlocal simulations is the selection of model parameters.
In this work we propose an optimization-based approach to parameter
identification for fractional models with an optional truncation radius. We
formulate the inference problem as an optimal control problem where the
objective is to minimize the discrepancy between observed data and an
approximate solution of the model, and the control variables are the fractional
order and the truncation length. For the numerical solution of the minimization
problem we propose a gradient-based approach, where we enhance the numerical
performance by an approximation of the bilinear form of the state equation and
its derivative with respect to the fractional order. Several numerical tests in
one and two dimensions illustrate the theoretical results and show the
robustness and applicability of our method
A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions
A rigorous mathematical framework is provided for a substructuring-based
domain-decomposition approach for nonlocal problems that feature interactions
between points separated by a finite distance. Here, by substructuring it is
meant that a traditional geometric configuration for local partial differential
equation problems is used in which a computational domain is subdivided into
non-overlapping subdomains. In the nonlocal setting, this approach is
substructuring-based in the sense that those subdomains interact with
neighboring domains over interface regions having finite volume, in contrast to
the local PDE setting in which interfaces are lower dimensional manifolds
separating abutting subdomains. Key results include the equivalence between the
global, single-domain nonlocal problem and its multi-domain reformulation, both
at the continuous and discrete levels. These results provide the rigorous
foundation necessary for the development of efficient solution strategies for
nonlocal domain-decomposition methods.Comment: 27 pages, 3 figure
Convergence Analysis and Numerical Studies for Linearly Elastic Peridynamics with Dirichlet-Type Boundary Conditions
The nonlocal models of peridynamics have successfully predicted fractures and
deformations for a variety of materials. In contrast to local mechanics,
peridynamic boundary conditions must be defined on a finite volume region
outside the body. Therefore, theoretical and numerical challenges arise in
order to properly formulate Dirichlet-type nonlocal boundary conditions, while
connecting them to the local counterparts. While a careless imposition of local
boundary conditions leads to a smaller effective material stiffness close to
the boundary and an artificial softening of the material, several strategies
were proposed to avoid this unphysical surface effect.
In this work, we study convergence of solutions to nonlocal state-based
linear elastic model to their local counterparts as the interaction horizon
vanishes, under different formulations and smoothness assumptions for nonlocal
Dirichlet-type boundary conditions. Our results provide explicit rates of
convergence that are sensitive to the compatibility of the nonlocal boundary
data and the extension of the solution for the local model. In particular,
under appropriate assumptions, constant extensions yield order
convergence rates and linear extensions yield order convergence
rates. With smooth extensions, these rates are improved to quadratic
convergence. We illustrate the theory for any dimension and
numerically verify the convergence rates with a number of two dimensional
benchmarks, including linear patch tests, manufactured solutions, and domains
with curvilinear surfaces. Numerical results show a first order convergence for
constant extensions and second order convergence for linear extensions, which
suggests a possible room of improvement in the future convergence analysis