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SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning
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Proposes SymMaP, an RNN-based risk-seeking deep symbolic discovery framework that learns interpretable closed-form expressions predicting optimal preconditioner parameters (SOR/SSOR omega, AMG threshold) from PDE coefficients, deployable in CPU solvers like PETSc.
Problem
Choosing preconditioning parameters (e.g., SOR relaxation factor, AMG threshold) for linear systems is instance-dependent; fixed defaults are suboptimal, while ML predictors are costly on CPUs and opaque.
Approach
Use risk-seeking policy gradient over an LSTM that generates prefix-notation symbolic expressions mapping PDE coefficient features to grid-searched optimal preconditioning parameters, yielding compact formulas that run with negligible overhead inside CPU solvers.
Key results
- For SOR, SymMaP reduces computation time by up to 40% vs PETSc default omega=1 and about 10% vs optimal constant (e.g., Biharmonic 1.24 s vs 2.04 s default and 1.31 s optimal constant).
- For SSOR, SymMaP cuts computation time by up to 27% vs PETSc default and 11% vs optimal constant.
- For AMG, SymMaP lowers the condition number by up to 40% vs PETSc default and 32% vs optimal constant; learned expression runtime is about 20% of an MLP on CPU with similar solve times.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SymMaP 1 instead of the optimal fixed constant for SOR | SOR · Biharmonic | computation time (seconds): 1.31 -> 1.24 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SOR | SOR · Darcy Flow | computation time (seconds): 9.54 -> 8.50 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SOR | SOR · Elliptic PDE | computation time (seconds): 16.6 -> 15.8 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SOR | SOR · Poisson | computation time (seconds): 1.38$\times 10^{-2}$ -> 1.35$\times 10^{-2}$ | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SOR | SOR · Thermal | computation time (seconds): 5.94$\times 10^{-2}$ -> 5.76$\times 10^{-2}$ | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SSOR | SSOR · Darcy Flow | computation time (seconds): 0.448 -> 0.412 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SSOR | SSOR · Elliptic PDE | computation time (seconds): 8.68 -> 7.70 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SSOR | SSOR · Poisson | computation time (seconds): 9.89$\times 10^{-3}$ -> 9.10$\times 10^{-3}$ | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for SSOR | SSOR · Thermal | computation time (seconds): 2.24$\times 10^{-2}$ -> 2.13$\times 10^{-2}$ | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for AMG | AMG · Darcy Flow | condition number: 7184 -> 4824 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for AMG | AMG · Elliptic PDE | condition number: 182.8 -> 168.8 | Helped |
| Use SymMaP 1 instead of the optimal fixed constant for AMG | AMG · Poisson | condition number: 4.55 -> 3.72 | Helped |
Surprising findings
- Learned expressions often depend on only one or two input features, ignoring most coefficients, consistent with physical intuition about diagonal dominance.
Limitations
- Evaluated only on SOR, SSOR, and AMG parameters; extension to ILU/ICC not yet done.
- Relies on grid search to produce labels, with high dataset generation cost (up to 100 hours per dataset).
- Restricted to symmetric or specific PDE families; biharmonic matrices incompatible with SSOR and AMG.
Experimental setup
Datasets: Darcy flow (FDM, Chebyshev permeability) second-order elliptic PDE (FDM) biharmonic equation (FEniCS DG) Poisson equation (FDM) thermal steady-state equation
Benchmarks: Darcy flow second-order elliptic PDE biharmonic Poisson thermal
Metrics: average computation time (s) average condition number of preconditioned matrix
Baselines: PETSc default parameters fixed constant optimal constant MLP
Builds on: deep symbolic optimization PPO PETSc
Closest papers by meaning
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- Learning Sparse Approximate Inverse Preconditioners for Conjugate Gradient Solvers on GPUs (NeurIPS 2025 · similarity 0.85)
- Learning to Relax: Setting Solver Parameters Across a Sequence of Linear System Instances (ICLR 2024 · similarity 0.85)
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