Data-driven linear solver selection and performance tuning for multiphysics simulations in porous media

TL;DR

The paper tackles the challenge of selecting and tuning preconditioned linear solvers for large, nonlinear multiphysics simulations in porous media. It introduces a data-driven solver selection framework based on a two-stage machine learning pipeline (classifier and regressor) that learns online from run-time performance data, updating incrementally as the simulation progresses. Across two model problems—coupled flow and heat transfer, and thermo-poromechanics with fractures—the method achieves robust solver choices with negligible overhead and near-optimal performance compared to an oracle with full historical data. This approach offers practical benefits for engineers and researchers lacking deep expert knowledge in linear solver tuning, enabling efficient and resilient simulations at industrial scales.

Abstract

Modeling multiphysics processes in porous media requires preconditioned iterative linear solvers to enable efficient simulations at industry-relevant scales. These solvers are typically composed of sub-algorithms that target individual physical processes. Various options are available for each algorithm, with the corresponding ranges of numerical parameters. The choices of sub-algorithms and their parameters significantly affects simulation performance and robustness. Optimizing these choices for each simulation is challenging due to the vast number of possible combinations. Moreover, optimization relies on performance data from past simulations, which becomes less representative as the simulation setup changes. This paper addresses the problem of automated selection and tuning of preconditioned linear solvers for multiphysics simulations. The proposed solver selection algorithm collects performance data during the run of the target simulation and continuously updates a machine learning model responsible for solver selection, resulting in an adaptively refined selection policy. The algorithm is evaluated on two time-dependent nonlinear model problems: (i) coupled fluid flow and heat transfer in porous media and (ii) thermo-poromechanics in porous media with fractures, governed by frictional contact mechanics. These experiments demonstrate that the algorithm selects efficient and robust solvers with negligible overhead and performs comparably to a reference selection policy that has full access to the performance data of prior simulations. Our results indicate that the proposed approach effectively addresses the challenge of solver selection and tuning, providing particular value to simulation engineers and researchers, especially when expert knowledge on linear solver tuning is not readily available.

Figures (16)

• Figure 1: The flowchart represents the solver configuration space for the coupled flow and transport problem. Diamond blocks correspond to algorithmic choices, oval blocks represent numerical parameters to tune, and rectangular blocks denote algorithms to choose from. Dotted arrows indicate that a single path must be chosen, while solid arrows connect components of a single linear solver configuration.
• Figure 2: Two solver configurations taken from the solver configuration space illustrated in \ref{['fig:solver_space']}. Configuration A represents a direct solver, while Configuration B represents GMRES, preconditioned by the CPR method.
• Figure 3: Machine learning pipeline application. In this example, the solver configuration space consists of 6 configurations. First, the classifier model discards solver configurations predicted to fail and retains only those expected to succeed. Then, the regression model predicts the rewards for those configurations. The configuration with the largest predicted reward is selected. Both the classifier and the regression models use the same solver encoding and simulation context as input.
• Figure 4: One of the simulations in Sequence A, the simulation time is 150 days. Left: permeability field slice. Right: cold temperature propagating from the injection well in the domain center, showing only temperature values below the threshold of 375 K.
• Figure 5: One of the simulations in Sequence B at the end of the simulation time. Left: two clusters of three fractures each. Right: cold temperature propagating from the injection well in the left fracture cluster.