Implicit Neural Solver for Time-dependent Linear PDEs with Convergence Guarantee
This work addresses the need for efficient PDE solvers in fields like physics and engineering, but it is incremental as it modifies existing semi-implicit methods with neural networks.
The authors tackled the problem of solving time-dependent linear PDEs by proposing a neural solver that learns an optimal iterative scheme, achieving faster convergence compared to semi-implicit schemes while preserving correctness and convergence guarantees.
Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for better stability and correctness. The existing implicit schemes are usually iterative and employ a general-purpose solver which may be sub-optimal for a specific class of PDEs. In this paper, we propose a neural solver to learn an optimal iterative scheme for a class of PDEs, in a data-driven fashion. We attain this objective by modifying an iteration of an existing semi-implicit solver using a deep neural network. Further, we prove theoretically that our approach preserves the correctness and convergence guarantees provided by the existing iterative-solvers. We also demonstrate that our model generalizes to a different parameter setting than the one seen during training and achieves faster convergence compared to the semi-implicit schemes.