arXiv subjects

Brittany A. Erickson

Publications and source records attributed to Brittany A. Erickson.

4 recordsLinked to original sources

arXiv2026

Exact Boundary Enforcement Along Implicit Geometries for Physics-Informed, Deep Learning Problems in Continuum Mechanics

Solutions to well-posed problems in continuum mechanics are continuously dependent upon prescribed boundary conditions. Because of this, variations in the enforcement of boundary data can impact the reliability of inversion techniques that rely on efficient and accurate forward models. To this end, it is necessary to understand how specific boundary implementation techniques can affect the performance of a given forward model. Our work focuses on the impact that key modeling decisions have on physics-informed neural network (PINN) solutions for initial boundary value problems in continuum mechanics. By interpolating boundary data over implicit boundary representations, we measure the performance of a physics-informed neural network across different configurations of soft and hard boundary enforcement. We target the problem of elastodynamic plane-strain and present a method of hard-enforcement of traction conditions over arbitrary, implicitly-defined, domain boundaries considering both first and second order formulations of the governing equations. We show that PINNs achieve a higher relative accuracy when solving the first-order plane strain problem and we observe a tradeoff between the final relative error and the total run time to complete training. This tradeoff is characterized by the number of hard and soft boundaries where, in the extremes, all soft-enforcement results in greater accuracy with a longer run time, while all hard-enforcement leads to lesser accuracy and a shorter run time.

physics.comp-ph↗

arXiv2025

Approximating Analytic Spectra of Hyperbolic Systems with Summation-by-Parts Finite Difference Operators

In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to accurately discover sources of physical instabilities. By considering the perturbed equations that arise in linearized problems, we study systems in which a lower-order term can act as a source of internal energy within the system. We apply high-order accurate summation-by-parts finite difference operators, with weak enforcement of boundary conditions through the simultaneous-approximation-term technique, which leads to a provably stable numerical discretization with formal order of accuracy given by $p = 2, 3, 4$ and $5$. We derive analytic solutions using Laplace transform methods, which provide important ground truth to ensure numerical convergence at the correct theoretical rate. We derive the analytic spectrum and find that it is better captured with mesh refinement, although dissipative strict stability (where the growth rate of the discrete problem is bounded above by the analytic) is not obtained. We also find that sole reliance on mesh refinement can be a problematic means for determining physical growth rates as some eigenvalues emerge (and persist with mesh refinement) based on spatial order of accuracy but are non-physical. We suggest that numerical methods be used to approximate the spectra when numerical stability is guaranteed and convergence of the numerical spectra is evident with both mesh refinement and increasing order of accuracy.

math.NA↗

arXiv2023

Physics-Informed Deep Learning of Rate-and-State Fault Friction

Direct observations of earthquake nucleation and propagation are few and yet the next decade will likely see an unprecedented increase in indirect, surface observations that must be integrated into modeling efforts. Machine learning (ML) excels in the presence of large data and is an actively growing field in seismology. However, not all ML methods incorporate rigorous physics, and purely data-driven models can predict physically unrealistic outcomes due to observational bias or extrapolation. Our work focuses on the recently emergent Physics-Informed Neural Network (PINN), which seamlessly integrates data while ensuring that model outcomes satisfy rigorous physical constraints. In this work we develop a multi-network PINN for both the forward problem as well as for direct inversion of nonlinear fault friction parameters, constrained by the physics of motion in the solid Earth, which have direct implications for assessing seismic hazard. We present the computational PINN framework for strike-slip faults in 1D and 2D subject to rate-and-state friction. Initial and boundary conditions define the data on which the PINN is trained. While the PINN is capable of approximating the solution to the governing equations to low-errors, our primary interest lies in the network's capacity to infer friction parameters during the training loop. We find that the network for the parameter inversion at the fault performs much better than the network for material displacements to which it is coupled. Additional training iterations and model tuning resolves this discrepancy, enabling a robust surrogate model for solving both forward and inverse problems relevant to seismic faulting.

math-ph↗

arXiv2021

Hybridized Summation-By-Parts Finite Difference Methods

We present a hybridization technique for summation-by-parts finite difference methods with weak enforcement of interface and boundary conditions for second order, linear elliptic partial differential equations. The method is based on techniques from the hybridized discontinuous Galerkin literature where local and global problems are defined for the volume and trace grid points, respectively. By using a Schur complement technique the volume points can be eliminated, which drastically reduces the system size. We derive both the local and global problems, and show that the linear systems that must be solved are symmetric positive definite. The theoretical stability results are confirmed with numerical experiments as is the accuracy of the method.

math.NA↗