PDE-NetGen 1.0: from symbolic partial differential equation (PDE) representations of physical processes to trainable neural network representations

Bridging physics and deep learning is a topical challenge. While deep learning frameworks open avenues in physical science, the design of physically consistent deep neural network architectures is an open issue. In the spirit of physics-informed neural networks (NNs), the PDE-NetGen package provides new means to automatically translate physical equations, given as partial differential equations (PDEs), into neural network architectures. PDE-NetGen combines symbolic calculus and a neural network generator. The latter exploits NN-based implementations of PDE solvers using Keras. With some knowledge of a problem, PDE-NetGen is a plug-and-play tool to generate physics-informed NN architectures. They provide computationally efficient yet compact representations to address a variety of issues, including, among others, adjoint derivation, model calibration, forecasting and data assimilation as well as uncertainty quantification. As an illustration, the workflow is first presented for the 2D diffusion equation, then applied to the data-driven and physics-informed identification of uncertainty dynamics for the Burgers equation.

Scalability of high-performance PDE solvers

Performance tests and analyses are critical to effective high-performance computing software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this article, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for partial differential equations (PDEs) that govern a wide range of physical applications. We consider a sequence of PDE-motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.

Tsunami Generation due to Supershear Earthquakes: A Case Study

<p>The 28 September 2018 Mw 7.5 Sulawesi strike-slip earthquake generated an unexpected tsunami with devastating consequences. Since such strike-slip earthquakes are not expected to generate large tsunamis, the latter’s origin remains much debated. A key notable feature of this earthquake is that it ruptured at supershear speed, i.e., with a rupture speed greater than the shear wave speed of the host medium. Dunham and Bhat (2008) showed that such supershear ruptures, in half-space, produce two shock fronts (or Mach fronts) corresponding to an exceedance of shear and Rayleigh wave speeds. The Rayleigh Mach front carries significant vertical velocity along its front. We couple the ground motion produced by such a supershear earthquake to a 1D non-linear shallow water wave equation that accounts for both the time-dependent bathymetric displacement as well its velocity. We use an extension of Fourier-based PDE solvers called the Fourier Continuation (FC) method to numerically solve the system. The FC method enables high-order convergence of Fourier series approximations of non-periodic functions by resolving the well-known Gibbs “ringing” effect.  FC-based solvers offer limited numerical dispersion, high-order accuracy and mild CFL conditions—making them ideal to solve this system. Using the local bathymetric profile of Palu bay around the Pantoloan harbor tidal gauge, we have been able to clearly reproduce the observed tsunami with minimal tuning of parameters. We conclude that the Rayleigh Mach front, generated by a supershear earthquake combined with the Palu bay geometry, caused the tsunami.</p>

PDE-NetGen 1.0: from symbolic PDE representations of physical processes to trainable neural network representations

Bridging physics and deep learning is a topical challenge. While deep learning frameworks open avenues in physical science, the design of physically-consistent deep neural network architectures is an open issue. In the spirit of physics-informed NNs, PDE-NetGen package provides new means to automatically translate physical equations, given as PDEs, into neural network architectures. PDE-NetGen combines symbolic calculus and a neural network generator. The later exploits NN-based implementations of PDE solvers using Keras. With some knowledge of a problem, PDE-NetGen is a plug-and-play tool to generate physics-informed NN architectures. They provide computationally-efficient yet compact representations to address a variety of issues, including among others adjoint derivation, model calibration, forecasting, data assimilation as well as uncertainty quantification. As an illustration, the workflow is first presented for the 2D diffusion equation, then applied to the data-driven and physics-informed identification of uncertainty dynamics for the Burgers equation.

Compression Challenges in Large Scale Partial Differential Equation Solvers

Solvers for partial differential equations (PDEs) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that need to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to the relatively small arithmetic intensity, and increasingly due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers over the recent decades. This paper surveys data compression challenges and discusses examples of corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to the main memory. We illustrate concepts for particular methods, with examples, and give references to alternatives.