arXiv subjects

Justin Dong

Publications and source records attributed to Justin Dong.

3 recordsLinked to original sources

arXiv2026

Leveraging higher-order time integration methods for improved computational efficiency in a rainshaft model

Cloud and precipitation microphysics packages in atmospheric general circulation models typically use first-order time integration methods with a large time step, requiring ad hoc limiters and substepping of the sedimentation scheme to prevent numerical instability. We investigate alternative methods for rain microphysical processes in the current Energy Exascale Earth System Model (E3SMv3), provided by the Predicted Particle Properties (P3) scheme. Using an offline rainshaft model containing P3's rain microphysics as a proxy for E3SMv3, we find that rain microphysics is underresolved in time at E3SMv3's default 300s time step. Accurately resolving the rainshaft model processes in time requires an over 100x reduction in time step, increasing wall clock time by 40x. However, higher-order time integrators based on Runge-Kutta methods offer improved solution accuracy for a given computational cost. Adaptive time stepping is key to obtaining computationally efficient microphysics results, eliminating the need for specialized substepping procedures in the sedimentation process. An adaptive second-order Runge-Kutta method approximates a high-temporal-resolution reference solution at only 2.6x the cost of the default P3 scheme; this method achieves high accuracy greater than 15x more efficiently than simply reducing the time step of P3 in the rainshaft model. We also analyze the timescales of the rain processes to obtain insight about the maximum time step each process is able to take while maintaining stability and accuracy, and about how individual processes should be grouped together for most efficient results.

physics.ao-ph↗

arXiv2024

Extended Galerkin neural network approximation of singular variational problems with error control

We present extended Galerkin neural networks (xGNN), a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs (2) an ``extended'' feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Numerical results are presented for several problems including steady Stokes flow around re-entrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

math.NA↗

arXiv2021

Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks. The finite-dimensional subspaces are then used to define a standard Galerkin approximation of the variational equation. This approach enjoys a number of advantages, including: the sequential nature of the algorithm offers a systematic approach to enhancing the accuracy of a given approximation; the sequential enhancements provide a useful indicator for the error that can be used as a criterion for terminating the sequential updates; the basic approach is largely oblivious to the nature of the partial differential equation under consideration; and, some basic theoretical results are presented regarding the convergence (or otherwise) of the method which are used to formulate basic guidelines for applying the method.

cs.LG↗