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Automated solver selection for simulation of multiphysics processes in porous media

Yury Zabegaev, Eirik Keilegavlen, Einar Iversen, Inga Berre

TL;DR

This work tackles the costly problem of solving strongly coupled multiphysics systems in porous media by framing solver selection as an online, context-aware, black-box optimization task. It develops a dual-branch methodology: Bayesian optimization with Gaussian Processes and a fast heuristic, both operating online at each time step to pick among a rich solver configuration space that includes splitting strategies, preconditioners, and numerical parameters. The framework accounts for dynamic regime shifts (e.g., diffusion vs advection dominance) by adapting solver choices in real time and leveraging knowledge transfer across similar simulations. Across poromechanics and non-isothermal flow tests, the approach reduces linear solve time relative to fixed baselines, with Gaussian Process-based methods providing the strongest performance at the expense of higher optimization overhead. The proposed framework is lightweight to embed in existing tools and supports expanding solver spaces, enabling accelerated exploration of new algorithms with reduced risk.

Abstract

Porous media processes involve various physical phenomena such as mechanical deformation, transport, and fluid flow. Accurate simulations must capture the strong couplings between these phenomena. Choosing an efficient solver for the multiphysics problem usually entails the decoupling into subproblems related to separate physical phenomena. Then, the suitable solvers for each subproblem and the iteration scheme must be chosen. The wide range of options for the solver components makes finding the optimum difficult and time-consuming; moreover, solvers come with numerical parameters that need to be optimized. As a further complication, the solver performance may depend on the physical regime of the simulation model, which may vary with time. Switching a solver with respect to the dominant process can be beneficial, but the threshold of when to switch solver is unclear and complicated to analyze. We address this challenge by developing a machine learning framework that automatically searches for the optimal solver for a given multiphysics simulation setup, based on statistical data from previously solved problems. For a series of problems, exemplified by successive time steps in a time-dependent simulation, the framework updates and improves its decision model online during the simulation. We show how it outperforms preselected state-of-the-art solvers for test problem setups. The examples are based on simulations of poromechanics and simulations of flow and transport. For the quasi-static linear Biot model, we demonstrate automated tuning of numerical solver parameters by showing how the L-parameter of the so-called Fixed-Stress preconditioner can be optimized. Motivated by a test example where the main heat transfer mechanism changes between convection and diffusion, we discuss how the solver selector can dynamically switch solvers when the dominant physical phenomenon changes with time.

Automated solver selection for simulation of multiphysics processes in porous media

TL;DR

This work tackles the costly problem of solving strongly coupled multiphysics systems in porous media by framing solver selection as an online, context-aware, black-box optimization task. It develops a dual-branch methodology: Bayesian optimization with Gaussian Processes and a fast heuristic, both operating online at each time step to pick among a rich solver configuration space that includes splitting strategies, preconditioners, and numerical parameters. The framework accounts for dynamic regime shifts (e.g., diffusion vs advection dominance) by adapting solver choices in real time and leveraging knowledge transfer across similar simulations. Across poromechanics and non-isothermal flow tests, the approach reduces linear solve time relative to fixed baselines, with Gaussian Process-based methods providing the strongest performance at the expense of higher optimization overhead. The proposed framework is lightweight to embed in existing tools and supports expanding solver spaces, enabling accelerated exploration of new algorithms with reduced risk.

Abstract

Porous media processes involve various physical phenomena such as mechanical deformation, transport, and fluid flow. Accurate simulations must capture the strong couplings between these phenomena. Choosing an efficient solver for the multiphysics problem usually entails the decoupling into subproblems related to separate physical phenomena. Then, the suitable solvers for each subproblem and the iteration scheme must be chosen. The wide range of options for the solver components makes finding the optimum difficult and time-consuming; moreover, solvers come with numerical parameters that need to be optimized. As a further complication, the solver performance may depend on the physical regime of the simulation model, which may vary with time. Switching a solver with respect to the dominant process can be beneficial, but the threshold of when to switch solver is unclear and complicated to analyze. We address this challenge by developing a machine learning framework that automatically searches for the optimal solver for a given multiphysics simulation setup, based on statistical data from previously solved problems. For a series of problems, exemplified by successive time steps in a time-dependent simulation, the framework updates and improves its decision model online during the simulation. We show how it outperforms preselected state-of-the-art solvers for test problem setups. The examples are based on simulations of poromechanics and simulations of flow and transport. For the quasi-static linear Biot model, we demonstrate automated tuning of numerical solver parameters by showing how the L-parameter of the so-called Fixed-Stress preconditioner can be optimized. Motivated by a test example where the main heat transfer mechanism changes between convection and diffusion, we discuss how the solver selector can dynamically switch solvers when the dominant physical phenomenon changes with time.
Paper Structure (23 sections, 36 equations, 8 figures, 4 tables)

This paper contains 23 sections, 36 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. The solver configuration space
  4. The context space
  5. Optimization problem
  6. Solver selection framework
  7. Online learning
  8. Machine learning methods
  9. Bayesian optimization approach
  10. Heuristic approach
  11. Model problems
  12. Poromechanical deformation
  13. Non-isothermal flow in porous media
  14. Numerical experiments
  15. Proving the concept of automated solver selection
  16. ...and 8 more sections

Figures (8)

  • Figure 1: (a) Example solver space. The hatched blocks represent categorical decisions. (b) The seven solver configurations.
  • Figure 2: Optimization of numerical solver configuration parameter for the Mandel's problem. (a) Variation of number of GMRES iterations with the stabilization coefficient $L$. (b) Evolution in $L$ from the solver selection framework during one simulation.
  • Figure 3: Cumulative performance of solver configurations picked by the solver selector compared to pre-selected configurations and random decisions. Lower is better. The bold line of each color displays the mean value, while the lighter-shaded region of the same color around it indicates the range from the lowest to highest value among all the repeats of an experiment.
  • Figure 4: Solver selection for the non-isothermal flow problem. Different colors represent the solver which was used for each time step.
  • Figure 5: Histograms of the invocations of different solver configurations. Average values and standard deviations are taken w.r.t. 20 repeats of the simulation. The numbering of the options corresponds to that given in Table \ref{['tab:extende_solver_options']}.
  • ...and 3 more figures