Jed Brown

Michael McCabe, Peter Harrington, Shashank Subramanian et al.

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to neural operators leads to significantly lower errors for long-term forecasts as well as longer time horizons without qualitative signs of divergence compared to the original models for these systems. We open-source our \href{https://github.com/mikemccabe210/stabilizing_neural_operators}{code} for reproducibility.

Shrirang Abhyankar, Jed Brown, Emil M. Constantinescu et al.

High-quality ordinary differential equation (ODE) solver libraries have a long history, going back to the 1970s. Over the past several years we have implemented, on top of the PETSc linear and nonlinear solver package, a new general-purpose, extensive, extensible library for solving ODEs and differential algebraic equations (DAEs). Package includes support for both forward and adjoint sensitivities that can be easily utilized by the TAO optimization package, which is also part of PETSc. The ODE/DAE integrator library strives to be highly scalable but also to deliver high efficiency for modest-sized problems. The library includes explicit solvers, implicit solvers, and a collection of implicit-explicit solvers, all with a common user interface and runtime selection of solver types, adaptive error control, and monitoring of solution progress. The library also offers enormous flexibility in selection of nonlinear and linear solvers, including the entire suite of PETSc iterative solvers, as well as several parallel direct solvers.

Mark Adams, Satish Balay, Oana Marin et al.

The communities who develop and support open source scientific software packages are crucial to the utility and success of such packages. Moreover, these communities form an important part of the human infrastructure that enables scientific progress. This paper discusses aspects of the PETSc (Portable Extensible Toolkit for Scientific Computation) community, its organization, and technical approaches that enable community members to help each other efficiently.

Michael McCabe, Jed Brown

The accuracy of simulation-based forecasting in chaotic systems is heavily dependent on high-quality estimates of the system state at the time the forecast is initialized. Data assimilation methods are used to infer these initial conditions by systematically combining noisy, incomplete observations and numerical models of system dynamics to produce effective estimation schemes. We introduce amortized assimilation, a framework for learning to assimilate in dynamical systems from sequences of noisy observations with no need for ground truth data. We motivate the framework by extending powerful results from self-supervised denoising to the dynamical systems setting through the use of differentiable simulation. Experimental results across several benchmark systems highlight the improved effectiveness of our approach over widely-used data assimilation methods.

Tristan Konolige, Jed Brown

Bundle adjustment is an important global optimization step in many structure from motion pipelines. Performance is dependent on the speed of the linear solver used to compute steps towards the optimum. For large problems, the current state of the art scales superlinearly with the number of cameras in the problem. We investigate the conditioning of global bundle adjustment problems as the number of images increases in different regimes and fundamental consequences in terms of superlinear scaling of the current state of the art methods. We present an unsmoothed aggregation multigrid preconditioner that accurately represents the global modes that underlie poor scaling of existing methods and demonstrate solves of up to 13 times faster than the state of the art on large, challenging problem sets.

Mark F. Adams, Jed Brown, Matt Knepley et al.

We investigate a domain decomposed multigrid technique, segmental refinement, for solving general nonlinear elliptic boundary value problems. Brandt and Diskin first proposed this method in 1994; we continue this work by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid can be eliminated on fine grids with modest amounts of extra work and storage while maintaining the asymptotic exactness of full multigrid, although we observe a dependence on an additional parameter not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores of a Cray XC30.

Jed Brown, Matthew G. Knepley, Barry F. Smith

Build-time configuration and environment assumptions are hampering progress and usability in scientific software. That which would be utterly unacceptable in non-scientific software somehow passes for the norm in scientific packages. The community needs reusable software packages that are easy use and flexible enough to accommodate next-generation simulation and analysis demands.