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We developed a new approach in converging equation-based modeling and data-driven learning on high-performance computing resources to accelerate physics-based earthquake simulations. Here, data-driven learning based on data generated while conducting equation-based modeling was used to accelerate the convergence process of an implicit low-order unstructured finite-element solver. This process involved a suitable combination of data-driven learning for estimating high-frequency components and coarsened equation-based models for estimating low-frequency components of the problem. The developed solver achieved a 12.8-fold speedup over the state-of-art solver with a 96.4% size-up scalability up to 24,576 nodes (98,304 MPI processes × 12 OpenMP threads = 1,179,648 CPU cores) of Fugaku with 126,581,788,413 degrees-of-freedom, leading to solving a huge city earthquake shaking analysis in a 10.1-fold shorter time than the previous state-of-the-art solver. Furthermore, to show that the developed method attains high performance on variety of systems with small implementation costs, we ported the developed method to recent GPU systems by use of directive based methods (OpenACC). The equation based modeling and the data-driven learning are of utterly different characteristics, and hence they are rarely combined. The developed approach of combining them is effective, and remarkable results mentioned above are achieved.

A low-order implicit unstructured finite-element solver for Earth's crust deformation problem is proposed. The solver scales up to 294,912 CPU cores of K computer with 94.8% size-up efficiency, enabling solving a 504 billion degrees-of-freedom crust deformation problem with 126 billion second-order tetrahedral elements in 146.5 s at 0.970 PFLOPS (20.6% of peak) on 294,912 CPU cores. This is 11.1 times faster than a highly-tuned preconditioned conjugate gradient solver and 2.18 times faster than the state-of-the-art SC14 Gordon Bell Prize Finalist solver. With the proposed solver, we solved a practical 381 billion degrees-of-freedom Eastern Japan crust deformation problem (576 km x 800 km x 400 km area; resolution of 62.5 m) in 252.2 s using 294,912 CPU cores. Such high-resolution analysis assures the convergence of stress and strain required to evaluate nonlinear material constitutive models at plate boundaries and is expected to become a key tool for physics-based earthquake forecasting.

Programming in HPC is a tedious work. Therefore functional programming languages that generate HPC programs have been proposed. However, they are not widely used by application scientists, because of learning barrier, and lack of demonstrated application performance. We have designed Formura which adopts application-friendly features such as typed rational array indices. Formura users can describe mathematical concepts such as operation over derivative operators using functional programming. Formura allows intuitive expression over array elements while ensuring the program is a stencil computation, so that state-of-the-art stencil optimization techniques such as temporal blocking is always applied to Formura-generated program. We demonstrate the usefulness of Formura by implementing a preliminary below-ground biology simulation. Optimized C-code are generated from 672 bytes of Formura program. The simulation was executed on the full nodes of the K computer, with 1.184 Pflops, 11.62% floating-point-instruction efficiency, and 31.26% memory throughput efficiency.

Real space DFT (RSDFT) is a simulation technique most suitable for massively-parallel architectures to perform first-principles electronic-structure calculations based on density functional theory. We here report unprecedented simulations on the electron states of silicon nanowires with up to 107,292 atoms carried out during the initial performance evaluation phase of the K computer being developed at RIKEN. The RSDFT code has been parallelized and optimized so as to make effective use of the various capabilities of the K computer. Simulation results for the self-consistent electron states of a silicon nanowire with 10,000 atoms were obtained in a run lasting about 24 hours and using 6,144 cores of the K computer. A 3.08 peta-flops sustained performance was measured for one iteration of the SCF calculation in a 107,292-atom Si nanowire calculation using 442,368 cores, which is 43.63% of the peak performance of 7.07 peta-flops.