Graph-grammar based algorithm for asteroid tsunami simulations

On January 18, 2022, around 1 million kilometers from Earth, five times the distance from Earth to the Moon, a large asteroid passed without harm to the Earth. Theoretically, however, the event of the asteroid falling into Earth, causing the tsunami, is possible since there are over 27,000 near-Earth asteroids [1], and the Earth’s surface is covered in 71 percent by water. We introduce a novel graph-grammar-based framework for asteroid tsunami simulations. Our framework adaptively generates the computational mesh of the Earth model. It is built from triangular elements representing the seashore and the seabed. The computational mesh is represented as a graph, with graph vertices representing the computational mesh element’s interiors and edges. Mesh refinements are often performed by the longest-edge refinement algorithm. We have expressed this algorithm by only two graph-grammar productions. The resulting graph represents the terrain approximating the topography with a prescribed accuracy. We generalize the graph-grammar mesh refinement algorithm to work on the entire Earth model, allowing the generation of the terrain topography, including the seabed. Having the seashore and the seabed represented by a graph, we introduce the finite element method simulations of the tsunami wave propagation. We illustrate the framework with simulations of the disastrous asteroid falling into the Baltic sea

Robust Variational Physics-Informed Neural Networks

We introduce a Robust version of the Variational Physics-Informed Neural Networks (RVPINNs) to approximate the Partial Differential Equations (PDEs) solution. We start from a weak Petrov-Galerkin formulation of the problem, select a discrete test space, and define a quadratic loss functional as in VPINNs. Whereas in VPINNs the loss depends upon the selected basis functions of a given test space, herein we minimize a loss based on the residual in the discrete dual norm, which is independent of the test space's choice of test basis functions. We demonstrate that this loss is a reliable and efficient estimator of the true error in the energy norm. The proposed loss function requires computation of the Gram matrix inverse, similar to what occurs in traditional residual minimization methods. To validate our theoretical findings, we test the performance and robustness of our algorithm in several advection-dominated-diffusion problems in one spatial dimension. We conclude that RVPINNs is a robust method

Physics Informed Neural Network Code for 2D Transient Problems (PINN-2DT) Compatible with Google Colab

We present an open-source Physics Informed Neural Network environment for simulations of transient phenomena on two-dimensional rectangular domains, with the following features: (1) it is compatible with Google Colab which allows automatic execution on cloud environment; (2) it supports two dimensional time-dependent PDEs; (3) it provides simple interface for definition of the residual loss, boundary condition and initial loss, together with their weights; (4) it support Neumann and Dirichlet boundary conditions; (5) it allows for customizing the number of layers and neurons per layer, as well as for arbitrary activation function; (6) the learning rate and number of epochs are available as parameters; (7) it automatically differentiates PINN with respect to spatial and temporal variables; (8) it provides routines for plotting the convergence (with running average), initial conditions learnt, 2D and 3D snapshots from the simulation and movies (9) it includes a library of problems: (a) non-stationary heat transfer; (b) wave equation modeling a tsunami; (c) atmospheric simulations including thermal inversion; (d) tumor growth simulations.Comment: 21 pages, 13 figure