This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for training. As a result, accurate evaluation may be impractical, and we show that naive approximations at replacing these integrals with unbiased estimates lead to biased loss functions and solutions. To overcome this bias, we investigate three types of potential solutions: the deterministic sampling approaches, the double-sampling trick, and the delayed target method. We consider three classes of PDEs for benchmarking; one defining Poisson problems with singular charges and weak solutions of up to 10 dimensions, another involving weak solutions on electro-magnetic fields and a Maxwell equation, and a third one defining a Smoluchowski coagulation problem. Our numerical results confirm the existence of the aforementioned bias in practice and also show that our proposed delayed target approach can lead to accurate solutions with comparable quality to ones estimated with a large sample size integral. Our implementation is open-source and available at https://github.com/ehsansaleh/btspinn.

Domain decomposition methods (DDMs) are popular solvers for discretized systems of partial differential equations (PDEs), with one-level and multilevel variants. These solvers rely on several algorithmic and mathematical parameters, prescribing overlap, subdomain boundary conditions, and other properties of the DDM. While some work has been done on optimizing these parameters, it has mostly focused on the one-level setting or special cases such as structured-grid discretizations with regular subdomain construction. In this paper, we propose multigrid graph neural networks (MG-GNN), a novel GNN architecture for learning optimized parameters in two-level DDMs. We train MG-GNN using a new unsupervised loss function, enabling effective training on small problems that yields robust performance on unstructured grids that are orders of magnitude larger than those in the training set. We show that MG-GNN outperforms popular hierarchical graph network architectures for this optimization and that our proposed loss function is critical to achieving this improved performance.

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the \textit{optimal} construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. In this work, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured- and unstructured-grid problems.

Large sparse linear systems of equations are ubiquitous in science and engineering, such as those arising from discretizations of partial differential equations. Algebraic multigrid (AMG) methods are one of the most common methods of solving such linear systems, with an extensive body of underlying mathematical theory. A system of linear equations defines a graph on the set of unknowns and each level of a multigrid solver requires the selection of an appropriate coarse graph along with restriction and interpolation operators that map to and from the coarse representation. The efficiency of the multigrid solver depends critically on this selection and many selection methods have been developed over the years. Recently, it has been demonstrated that it is possible to directly learn the AMG interpolation and restriction operators, given a coarse graph selection. In this paper, we consider the complementary problem of learning to coarsen graphs for a multigrid solver, a necessary step in developing fully learnable AMG methods. We propose a method using a reinforcement learning (RL) agent based on graph neural networks (GNNs), which can learn to perform graph coarsening on small planar training graphs and then be applied to unstructured large planar graphs, assuming bounded node degree. We demonstrate that this method can produce better coarse graphs than existing algorithms, even as the graph size increases and other properties of the graph are varied. We also propose an efficient inference procedure for performing graph coarsening that results in linear time complexity in graph size.

Parallel applications are often unable to take full advantage of emerging parallel architectures due to scaling limitations, which arise due to inter-process communication. Performance models are used to analyze the sources of communication costs. However, traditional models for point-to-point communication fail to capture the full cost of many irregular operations, such as sparse matrix methods. In this paper, a node-aware based model is presented. Furthermore, the model is extended to include communication queue search time as well as an additional parameter estimating network contention. The resulting model is applied to a variety of irregular communication patterns throughout matrix operations, displaying improved accuracy over traditional models.

With the rate of errors that can silently effect an application's state/output expected to increase on future HPC machines, numerous application-level detection and recovery schemes have been proposed. Recovery is more efficient when errors are contained and affect only part of the computation's state. Containment is usually achieved by verifying all information leaking out of a statically defined containment domain, which is an expensive procedure. Alternatively, error propagation can be analyzed to bound the domain that is affected by a detected error. This paper investigates how silent data corruption (SDC) due to soft errors propagates through three HPC applications: HPCCG, Jacobi, and CoMD. To allow for more detailed view of error propagation, the paper tracks propagation at the instruction and application variable level. The impact of detection latency on error propagation is shown along with an application's ability to recover. Finally, the impact of compiler optimizations are explored along with the impact of local problem size on error propagation.

This paper presents IPAS, an instruction duplication technique that protects scientific applications from silent data corruption (SDC) in their output. The motivation for IPAS is that, due to natural error masking, only a subset of SDC errors actually affects the output of scientific codes—we call these errors silent output corruption (SOC) errors. Thus applications require duplication only on code that, when affected by a fault, yields SOC. We use machine learning to learn code instructions that must be protected to avoid SOC, and, using a compiler, we protect only those vulnerable instructions by duplication, thus significantly reducing the overhead that is introduced by instruction duplication. In our experiments with five workloads, IPAS reduces the percentage of SOC by up to 90% with a slowdown that ranges between 1.04x and 1.35x, which corresponds to as much as 47% less slowdown than state-of-the-art instruction duplication techniques.

The "postal" model of communication [3, 8] T = α + βn, for sending n bytes of data between two processes with latency α and bandwidth 1/β, is perhaps the most commonly used communication performance model in parallel computing. This performance model is often used in developing and evaluating parallel algorithms in high-performance computing, and was an effective model when it was first proposed. Consequently, numerous tests of "ping pong" communication have been developed in order to measure these parameters in the model. However, with the advent of multicore nodes connected to a single (or a few) network interfaces, the model has become a poor match to modern hardware. In this paper, we show a simple three-parameter model that better captures the behavior of current parallel computing systems, and demonstrate its accuracy on several systems. In support of this model, which we call the max-rate model, we have developed an open source benchmark1 that can be used to determine the model parameters.

With the rate of errors that silently effect an application's state/output expected to increase in future HPC machines, numerous mitigation schemes have been proposed, but little work has been done investigating why these schemes detect some error while other is masked. This paper investigates how silent data corruption (SDC) propagates through a sparse matrix vector multiply (SpMV), a fundamental HPC computation kernel. We discover that analyzing the mathematics of the SpMV limits understanding of SDC propagation. We achieve a more complete understanding by investigating how SDC propagates in a SpMV as it is expressed in machine instructions.

Irregular computations on large workloads are a necessity in many areas of computational science. Mapping these computations to modern parallel architectures, such as GPUs, is particularly challenging because the performance often depends critically on the choice of data-structure and algorithm. In this paper, we develop a parallel processing scheme, based on Merge Path partitioning, to compute segmented row-wise operations on sparse matrices that exposes parallelism at the granularity of individual nonzero entries. Our decomposition achieves competitive performance across many diverse problems while maintaining predictable behaviour dependent only on the computational work and ameliorates the impact of irregularity. We evaluate the performance of three sparse kernels: Spiv, Spaded and Sperm. We show that our processing scheme for each kernel yields comparable performance to other schemes in many cases and our performance is highly correlated, nearly 1, to the computational work irrespective of the underlying structure of the matrices.

Sparse matrix--matrix multiplication (SpGEMM) is a key operation in numerous areas from information to the physical sciences. Implementing SpGEMM efficiently on throughput-oriented processors, such as the graphics processing unit (GPU), requires the programmer to expose substantial fine-grained parallelism while conserving the limited off-chip memory bandwidth. Balancing these concerns, we decompose the SpGEMM operation into three highly parallel phases: expansion, sorting, and contraction, and introduce a set of complementary bandwidth-saving performance optimizations. Our implementation is fully general and our optimization strategy adaptively processes the SpGEMM workload row-wise to substantially improve performance by decreasing the work complexity and utilizing the memory hierarchy more effectively.